Variations on Artin's Primitive Root Conjecture

Variations on Artin’s Primitive Root Conjecture

Adam Tyler Felix

A thesis submitted to the Department of Mathematics & Statistics in conformity with the requirements for the degree of Doctor of Philosophy

Queen’s University Kingston, Ontario, Canada August 2011

i Abstract

Let a ∈ Z be a non-zero integer. Let p be a prime such that p - a. Deﬁne the index

∗ of a modulo p, denoted ia(p), to be the integer ia(p) := [(Z/pZ) : ha mod pi]. Let

Na(x) := #{p ≤ x : ia(p) = 1}. In 1927, Emil Artin conjectured that

Na(x) ∼ A(a)π(x)

where A(a) > 0 is a constant dependent only on a and π(x) := {p ≤ x : p prime}.

Rewrite Na(x) as follows: X Na(x) = f(ia(p)), p≤x

where f : N → C with f(1) = 1 and f(n) = 0 for all n ≥ 2. We examine which other functions f : N → C will give us formulæ

X f(ia(p)) ∼ caπ(x), p≤x

where ca is a constant dependent only on a. Deﬁne ω(n) := #{p|n : p prime}, Ω(n) := #{d|n : d is a prime power} and

ii d(n) := {d|n : d ∈ N}. We will prove

X x (log(i (p)))α = c π(x) + O a a (log x)2−α−ε p≤x X x log log x ω(i (p)) = c0 π(x) + O a a (log x)2 p≤x X x log log x Ω(i (p)) = c00π(x) + O a a (log x)2 p≤x and X x d(i ) = c000π(x) + O a a (log x)2−ε p≤x for all ε > 0.

We also extend these results to ﬁnitely-generated subgroups of Q∗ and E(Q) where E is an elliptic curve deﬁned over Q.

iii Acknowledgments

First, I would like to thank my supervisor, Dr. Ram Murty, for his patience, exper- tise, and knowledge. He has a perspicacity in analytic number theory seen in only few individuals on this planet. The past ﬁve years have been enlightening. I would also like to thank the members of the defense examination for their com- ments: Drs. E. Kani, O. Bogoyavlenskij, A. Akbary, S. Akl, and R. Henriksen. I would also like to thank my alma mater, the University of Waterloo, for preparing me so well in mathematics. I would especially like to thank Dr. Yu-Ru Liu for intro- ducing me the analytic number theory, sieve theory, and having conﬁdence in me. I would also like to thank Drs. S. Furino and C. Struthers and Ms. Shaw for making mathematics challenging and enjoyable. Finally, I want to thank my family for their unconditional love and support. My father for showing a good work ethic will get you anywhere. My brother for providing humour whenever it was needed. My sister for always wanting to play games. My mother for everything, you have taught so much and I will never be able to repay you. Thank you and I love you all.

iv Statement of Originality

I hereby certify that all of the work described within this thesis is the original work of the author. Any published (or unpublished) ideas and/or techniques from the work of others are fully acknowledged in accordance with the standard referencing practices.

v Table of Contents

Abstract ii

Acknowledgments iv

Statement of Originality v

Table of Contents vi

1 Introduction 1 1.1 Artin’s Conjecture on Primitive Roots ...... 1 1.2 Generalizing Artin’s Conjecture ...... 3 1.3 Deﬁnitions and Notations ...... 6 1.3.1 Arithmetic Functions ...... 7 1.3.2 Asymptotic, Big O, and Little o Notation ...... 9 1.3.3 Elliptic Curves ...... 10 1.3.4 The GRH and the Dedekind Zeta Function ...... 11 1.4 Organization of the Thesis ...... 14

Chapter 2: A New Method ...... 16 2.1 Introduction ...... 16 2.2 Lemmata ...... 19 2.3 The Function (log n)α ...... 23 2.4 The Divisor Function ...... 26 2.4.1 The First Summation ...... 27 2.4.2 The Second Summation ...... 28 2.4.3 The Third Summation ...... 30 2.4.4 Computation of the Constant ...... 32

Chapter 3: Finitely-Generated Analogues ...... 40 3.1 Statement of Theorems ...... 40

vi 3.1.1 The r-rank Theorem ...... 41 3.1.2 An Immediate Corollary ...... 44 3.1.3 The Fixed Index Theorem ...... 46 3.2 An Asymptotic Formula ...... 47 3.3 The Fixed Index Theorem ...... 55

Chapter 4: Elliptic Curve Analogues ...... 61 4.1 Statement of Theorems ...... 63 4.2 A Divisibility Criterion ...... 67 4.3 An Asymptotic Result ...... 71 4.3.1 Small Primes ...... 72 4.3.2 Intermediate and Large Primes ...... 74 4.3.3 Optimal Error Terms ...... 76 4.4 Proofs of Theorems 4.1.2 and 4.1.3 ...... 79

Chapter 5: Summations related to the Artin’s Conjecture ...... 84 5.1 Generalizing the Titchmarsh Divisor Problem ...... 84 5.1.1 Application to a Generalization of Artin’s Conjecture . . . . . 89 5.2 The number of prime and prime power divisors of ia(p) ...... 94 5.3 Fomenko’s Result on Average ...... 99

Chapter 6: A Conjecture of Erd˝osand Fomenko’s Theorem . . . . . 105 6.1 Erd˝os’sConjecture ...... 105 6.2 Proof of Theorem 6.1.4 ...... 108

Chapter 7: Further Research ...... 111 7.1 A Second Conjecture of Fomenko ...... 111 P 7.2 Diﬃculties with p≤x log(ia(p)) ...... 114 7.3 Other Generalizations and Potential Improvements ...... 116 7.3.1 Elliptic Curves ...... 116 Towards a Chebotarev Condition ...... 117 7.3.2 Other Sequences and Functions ...... 118 7.4 Unconditional Lower Bounds ...... 120

Bibliography ...... 123

vii Chapter 1

Introduction

1.1 Artin’s Conjecture on Primitive Roots

Let p ∈ N be a prime number. Let us consider (Z/pZ)∗ := {a mod p : p - a}. It can be shown that (Z/pZ)∗ is cyclic. That is, there exists a ∈ Z such that (Z/pZ)∗ = ha mod pi. In this case we say that a is a primitive root modulo p. In fact, it can also be shown that the number of generators of (Z/pZ)∗ which have the form a mod p in (Z/pZ)∗ is ϕ(p − 1) where ϕ(n) = #{1 ≤ k ≤ n : gcd(k, n) = 1}. In [25], Gauß used primitive roots to discuss the periodicity of 1/p for p a prime not equal to 2 or 5. Both Euler and Jacobi used primitive roots before Gauß. In 1927, Emil Artin (see [43, Introduction] and [31]) reversed the above analysis into the following question: let a be a ﬁxed integer such that a 6= 0, ±1 or a perfect

h square. Let a = b where b ∈ Z is not a perfect power and h ∈ N. Deﬁne Na(x) := #{p ≤ x :(Z/pZ)∗ = ha mod pi}. Artin conjectured that the following is true:

Na(x) ∼ Ahπ(x)

1 CHAPTER 1. INTRODUCTION 2

where π(x) = #{p ≤ x : p prime} and Y 1 Y 1 A = 1 − 1 − > 0. (1.1) h q − 1 q(q − 1) q|h q-h q prime q prime The heuristic behind this conjecture is based on the following idea: we have a is primitive root modulo p if and only if for all q prime the following conditions do not occur:

p ≡ 1 (mod q)

p−1 a q ≡ 1 (mod p).

The “probability” the ﬁrst condition above holds is 1/ϕ(q) = 1/(q − 1). The “prob-

p−1 ability” the second condition above holds is 1/q since a q is a qth root of unity and there are exactly q of those. This conjecture is still unresolved. However, Hooley [31] has provided a condi- tional resolution to the problem:

Theorem 1.1.1 (Hooley). Suppose a ∈ Z such that a 6= 0, ±1 or a perfect square. Suppose further that the generalized Riemann hypothesis holds for Dedekind zeta func-

1/k th tions for the ﬁelds Q(a , ζk) with k ∈ N squarefree and where ζk is a k root of unity. Then, x log log x N (x) = A(a)π(x) + O a (log x)2 where the implied constant is dependent on a.

We will define Dedekind zeta functions, the generalized Riemann hypothesis, and O notation in a later section. It should be noted that A(a) in Theorem 1.1.1 is different from Ah in (1.1). It was initially discovered by Lehmer [45] that Artin’s original constant was off by a by small factor. In fact, let h be as above and let CHAPTER 1. INTRODUCTION 3

2 a = a1a2 where a1, a2 ∈ Z and a1 is squarefree. If a1 6≡ 1 (mod 4), then A(a) = Ah, and if a1 ≡ 1 (mod 4), then      Y 1 Y 1  A(a) = Ah 1 − µ(|a1|)  .  q − 2 q2 − q − 1  q| gcd(h,a1) q-h  q prime q|a1 q prime The best unconditional results are of the following flavour: One of 2, 3, or 5 is a primitive root modulo p for infinitely many primes p. In fact, we have cx #{p ≤ x : a is a primitive root modulo p} ≥ , (log x)2 where c > 0 is a constant, and a is one of 2, 3, or 5. This result originates in the work of Gupta and Murty [26], and was later refined by Heath-Brown [29].

1.2 Generalizing Artin’s Conjecture

Let a be as before, and let p be a prime such that p - a. Then, deﬁne the order of a

modulo p, denoted fa(p), as

k fa(p) := min{k ∈ N : a ≡ 1 mod p} = |ha mod pi|.

Since p - a, fa(p) is well-deﬁned by Fermat’s Little Theorem. Deﬁne the index of a

modulo p, denoted ia(p), as

∗ p − 1 ia(p) := [(Z/pZ) : ha mod pi] = . fa(p) We note that Gauß [25, Article 57] deﬁned the index of a modulo p with respect to

a primitive root g of (Z/pZ)∗ as follows: since p is a prime, there exists g ∈ Z such that (Z/pZ)∗ = hg mod pi. Then, the index of a modulo p with respect to g, denoted

p i indg(a), is the number i ∈ {1, 2, . . . , p − 1} such that g ≡ a mod p. It should be CHAPTER 1. INTRODUCTION 4

p noted that the two deﬁnitions are related by the fact that ia(p) = gcd(p − 1, indg(a)) for any primitive root modulo p. To see this we note that from group theory we have

p−1 p−1 |ha mod pi| = p . However, by the above deﬁnitions, |ha mod pi| = , gcd(p−1,indg(a)) ia(p) and so, the claim holds. Throughout, index will denote the index of a subgroup as

p opposed to indg(a). We can reformulate Artin’s conjecture in the following manner:

Na(x) = #{p ≤ x : fa(p) = p − 1}

= #{p ≤ x : ia(p) = 1} X = χ{1}(ia(p)) p≤x where for S ⊂ N,   0 if n∈ / S χS(n) = .  1 if n ∈ S

We would like to know what would occur if we change χ{1} to a generic function

f : N → C. That is, can we obtain the following relation X f(ia(p)) ∼ Af (a)π(x) p≤x

where Af (a) is a constant dependent on f and a? This question was ﬁrst studied by Stephens [67], then by Wagstaﬀ [72], Murata [48], Elliott and Murata [15], Pap- palardi [54], Bach, Lukes, Shallit, and Williams [3], and Fomenko [21] among others. Of course, the functions f will have reasonable restrictions so as to not force an im- possibility with the above relation. For example, f(x) = x does not satisfy the above relation. We will see this in Chapter 7. We will mostly be interested in the functions f(n) = log n, ω(n), Ω(n), and P d(n) := d|n 1 where ω(n) and Ω(n) are the number of primes dividing n counting CHAPTER 1. INTRODUCTION 5

without and with multiplicity, respectively. We will also be concerned with χ{w} (with w ∈ N ﬁxed) in a few sections. For f(x) = log x and a = 2, The authors of [3] made the following conjecture and provided computational evidence for its veracity: X log(i2(p)) ∼ c2li(x). p≤x They also mention that their heuristics should follow for a > 2 and give computational evidence for a = 3 and a = 5. Pappalardi [54] has shown the following theorem:

Theorem 1.2.1 (Pappalardi). Let a ∈ Z diﬀerent from 0 and ±1. We have x X x log log x log(i (p)) log x a log x p≤x where the lower bound is unconditional, and for the upper bound we suppose the generalized Riemann hypothesis holds for Dedekind zeta functions for ﬁelds of type

1/k th Q(a , ζk) where ζk is a k root of unity.

For any a an integer not equal to 0, ±1 and a a squarefree integer, we have the following theorem of Fomenko [21]:

Theorem 1.2.2 (Fomenko). Suppose the generalized Riemann hypothesis holds for

1/k th Dedekind zeta functions for ﬁelds of type Q(a , ζk) where ζk is a k root of unity. Suppose further Conjecture A of Hooley holds. Then, X x log log x log(i (p)) = c li(x) + O a a (log x)2 p≤x

where ca is an eﬀectively computable constant dependent on a, and Z x 1 li(x) := dt. 2 log t CHAPTER 1. INTRODUCTION 6

We note that this theorem can be extended to a 6= 0 or ±1. We will discuss Conjecture A of Hooley [32] in Chapter 3. In Chapter 7, we will see some of the diﬃculties in attempting to prove the asymptotic formula for f(x) = log x without the assumption that Conjecture A is true.

1.3 Deﬁnitions and Notations

Throughout this document, N, Z, Q, R, R+, and C will denote positive integers, integers, rational numbers, real numbers, positive real numbers, and complex numbers,

respectively. If S is a subset of C, then Q(S) will denote the smallest ﬁeld containing Q and S, which is inside C. In Chapter 4, we also allow S ⊂ R2. For these S, Q(S) will

denote the smallest field containing Q and real numbers x1, x2 for every (x1, x2) ∈ S, which is again inside C. Also p and q ∈ N will denote prime numbers unless otherwise specified, and a, d, k, `, m, n, and w will denote integers unless otherwise specified. In addition, summations of the form X X X X

p≤x qα≤x a≤x p≤x d|p−a will indicate the variables are over the primes, prime powers, integers, and primes with a speciﬁed condition where all symbols match the above conventions. The variables

x, y, and z will denote elements of R+ unless otherwise speciﬁed. For s ∈ C, <(s) will denote the real part of the complex number s.

R x −1 As already mentioned π(x) := #{p ≤ x}, li(x) := 2 (log t) dt, and for a, k ∈ Z, we deﬁne π(x; k, a) := #{p ≤ x : p ≡ a mod k}.

If g1, g2, . . . , gm ∈ G where G is a group, then hg1, g2, . . . , gmi will denote the

smallest subgroup of G containing g1, g2, . . . , gm. CHAPTER 1. INTRODUCTION 7

If F ⊂ K are fields, then [K : F ] is the dimension of K as an F -vector space. Let f(x) ∈ F [x], where F is a field. Let f(x) factor over the algebraic closure F or the splitting field K of f(x) as c(x − α1)(x − α2) ··· (x − αn) where n = deg(f), c ∈ F , and α1, α2, . . . , αn ∈ K. The discriminant of f(x) is the product

Y 2 (αi − αj) . i

As already deﬁned for p - a, we deﬁne fa(p) and ia(p) to be the order of a modulo p, and the index of a modulo p, respectively. We also say a is a primitive root modulo p if fa(p) = p − 1, or equivalently, ia(p) = 1.

Let a1, a2, . . . , ar ∈ Q. We say that a1, a2, . . . , ar are multiplicatively independent if and only if the equation

n1 n2 nr a1 a2 ··· ar = 1

r has exactly one solution (n1, n2, . . . , nr) ∈ Z that being (0, 0,..., 0). We note that this implies none of a1, a2, . . . , ar can be ±1 or 0.

1.3.1 Arithmetic Functions

We deﬁne the following arithmetic functions f : N → C: For n ∈ N, we deﬁne

ϕ(n) := #{a mod n : gcd(a, n) = 1}, X ω(n) := #{p|n} = 1, p|n

α X Ω(n) := #{p |n : α ∈ N} = α, pαkn CHAPTER 1. INTRODUCTION 8

where pαkn if and only if pα|n and pα+1 - n,   1 if n = 1   µ(n) := (−1)ω(n) if n is squarefree ,    0 otherwise X d(n) := #{d ∈ N : d|n} = 1, d|n

k dk(n) = #{(d1, d2, . . . , dk) ∈ N : n = d1d2 ··· dk},

where k ∈ N, and   α log p if n = p with α ∈ N Λ(n) := .  0 otherwise We call µ(n) the M¨obiusfunction, ϕ(n) the Euler phi or totient function, d(n) the divisor function, and Λ(n) the von Mangoldt function.

Recall the Fundamental Theorem of Arithmetic that every n ∈ N can be expressed

α1 α2 αk uniquely (up to ordering of p1, p2, . . . , pk) as p1 p2 ··· pk where p1, p2, . . . , pk ∈ N

are distinct prime numbers and α1, α2, . . . , αk ∈ N. Thus, for n ∈ N, we may write Y n = pνp(n) p

where νp(n) = αi if p = pi and 0 otherwise. We call νp the p-adic valuation. That

α is, νp(n) := max{α ∈ Z : α ≥ 0, p kn}. We can extend this deﬁnition to Q in the

following way: let a/b ∈ Q such that gcd(a, b) = 1. Then, νp(a/b) = νp(a) − νp(b). CHAPTER 1. INTRODUCTION 9

1.3.2 Asymptotic, Big O, and Little o Notation

Let A be an unbounded subset of R. Let f : A → C and g : A → R+ be two functions. We say that f is asymptotic to g if f(x) lim = 1. x→∞ g(x) x∈A In this case, we write f ∼ g. We write

f(x) = O(g(x)) or equivalently,

f(x) g(x)

if there exists a positive constant A such that |f(x)| ≤ Ag(x) for all x ∈ A. In this case, we say that f is big O of g. Also, we write g(x) f(x) to mean f(x) g(x), and we write f(x) g(x) if f(x) g(x) and g(x) f(x). Here we

require f : A → R+. We write

f(x) = o(g(x))

if f(x) lim = 0. x→∞ g(x) x∈A In this case, we say f(x) is little o of g.

If is one of the relations ∼, , , or f(x) = o(g(x)), then the notation f(x)yg(x) implies that the relation is dependent on y. For example, it is well known that d(n) nε for every ε > 0. However, the constant C for which d(n) ≤ Cnε for all

ε ε n ∈ N is dependent on ε. So we may also write d(n) ε n or d(n) = Oε(n ). CHAPTER 1. INTRODUCTION 10

1.3.3 Elliptic Curves

In Chapter 4, E will denote an elliptic curve deﬁned over Q. Since the characteristic of Q is not 2 or 3, E can be deﬁned as

2 3 E(Q) := {(x, y): x, y ∈ Q, y = x + Ax + B} ∪ {∞} for some A, B ∈ Q and where ∞ is a distinguished point given in projective coordinates by [0 : 1 : 0]. Now E(Q) is an Abelian group with ∞ being the identity and addition being defined using lines of intersection and addition. The equation y2 = x3 + Ax + B is called a Weierstraß equation defining E. In fact, for any field K with characteristic not equal to 2 or 3, there is an equation of this form. For characteristics 2 and 3, the Weierstraß equation is defined slightly differently (ee Appendix A of Silverman [63]). The discriminant of E is defined as the discriminant of a minimal Weierstraß equation x3 + Ax + B defining E, which exists by [63, Corollary VIII.8.3] since E is defined over Q. In fact, we have any equation y2 = x3 + Ax + B is an elliptic curve if and only if the discriminant of x3 + Ax + B is non-zero.

We say that a prime p is of good reduction if E deﬁned over Q by y2 = x3 + Ax + B is still an elliptic curve when A and B are reduced modulo p. That

2 3 is, when y = x + (A mod p)x + (B mod p) is also an elliptic curve deﬁned over Fp. All other primes are called primes of bad reduction. We note that there are only ﬁnitely many primes of bad reduction. These primes correspond to the prime divisors of the discriminant of E (see Silverman [63, Proposition VII.5.1]). CHAPTER 1. INTRODUCTION 11

1.3.4 The GRH and the Dedekind Zeta Function

Let us ﬁrst discuss the Riemann Hypothesis. For <(s) > 1, let −1 X 1 Y 1 ζ(s) := = 1 − . ns ps n≥1 p This function is called the Riemann-zeta function. Riemann [58] was able to show that ζ(s) could be analytically extended to the whole complex plane except for a simple pole at s = 1 with residue 1. Also, he was able to show the following functional equation holds for this analytically continued function: − s s − 1−s 1 − s π 2 Γ ζ(s) = π 2 Γ ζ(1 − s) 2 2 where Γ(z) is the complex analytic continuation of the function f : R → R given by R ∞ x−1 −t f(x) = 0 t e dt, valid for x > 0. 1 From this functional equation, one can easily see that the line <(s) = 2 is signif- icant in the study of ζ(s). At this point, Riemann hypothesized that for all s ∈ C with ζ(s) = 0 and <(s) > 0 we have <(s) = 1/2. This is known as the Riemann hypothesis. If it is true, then √ π(x) = li(x) + O x log x . (1.2)

Currently, the best known result of this nature is the following: there exists some constant c > 0 such that 3 !! c(log x) 5 π(x) = li(x) + O x exp − (log log x)1/5 which is clearly weaker than (1.2). This improvement in the classical error term of x exp(−c(log x)1/2) are originally due to N. M. Korobov [38] and I. M. Vinogradov [71], and later, H.-E. Richert [57]. CHAPTER 1. INTRODUCTION 12

Let K be an algebraic number ﬁeld. That is, Q ⊂ K and [K : Q] < ∞. Deﬁne

OK := {α ∈ K : f(α) = 0 for some monic f(x) ∈ Z[x]}.

It can be shown that OK is a ring. In fact, it can be shown that any ideal of OK can be written as a product of prime ideals in OK uniquely (up to ordering). For example, let p ∈ Z be a prime. Then, pOK is an ideal of OK . Thus, there exist prime ideals p1, p2,..., pg and positive integers e1, e2, . . . , eg such that

e1 e2 eg pOK = p1 p2 ··· pg .

Also, it can be shown that |OK /a| < ∞ for all non-zero ideals a in OK . In fact, for a prime ideal p with the property that p ∈ p and p ∈ Z is prime, it can be shown that

f |OK /p| = p for some f ∈ N (see [37, Chapter 1], [53, Chapter 1], or [18, §5.3]). Let

e1 e2 eg pOK = p1 p2 ··· pg .

fi and let |OK /pi| = p where fi ∈ N for all i ∈ {1, 2, . . . , g}. Then, we have g X eifi = [K : Q] i=1

(see [18, Exercise 5.3.17]). We say that p splits completely in K if ei = fi = 1 for all i ∈ {1, 2, . . . , g}. We say that p ramiﬁes in K if ei > 1 for some i ∈ {1, 2, . . . , g}.

For s ∈ C with <(s) > 1 deﬁne: X 1 ζ (s) := K N (a)s a K

where the summation is over non-zero ideals a in OK and NK (a) = |OK /a|. We call

ζK (s) the Dedekind-zeta function of K.

Hecke [30] showed that ζK (s) has an analytic continuation to the whole complex

plane except for a simple pole at s = 1. Let n = [K : Q], let σ1, σ2, . . . , σn : K → C

be the embeddings of K into C, and let ω1, ω2, . . . , ωn be a Z-basis for OK . Then, the CHAPTER 1. INTRODUCTION 13

discriminant of K, denoted by ∆K , is deﬁned as  2 σ1(ω1) σ1(ω2) . . . σ1(ωn)     σ2(ω1) σ2(ω2) . . . σ2(ωn) ∆ := det   . K  . . . .   . . .. .      σn(ω1) σn(ω2) . . . σn(ωn)

The conjugates of θ ∈ K are σ1(θ), σ2(θ), . . . , σn(θ). Hecke [30] also showed the following functional equation holds:

r −r s − [K:Q]s s s 1 r 2 2 π 2 |∆ | 2 Γ Γ(s) 2 ζ (s) K 2 K r1 −r (1−s) − [K:Q](1−s) 1−s 1 − s r = 2 2 π 2 |∆ | 2 Γ Γ(1 − s) 2 ζ (1 − s) K 2 K

∆K is the discriminant of K over Q and when we write K = Q(θ), then r1 is the

number of real conjugates of θ over Q and 2r2 is the number of complex conjugates of θ over Q. 1 Again, one can see that the line <(s) = 2 is important to the study of ζK (s). 1 Suppose for all s ∈ C with <(s) > 0 and ζK (s) = 0 we have <(s) = 2 . This is known as the generalized Riemann hypothesis for the Dedekind zeta function of

K which we will denote by GRH. Using the fact that primes in Z = OQ correspond to prime ideals in Z we define p to be prime in OK if p is a prime ideal of OK . Let K be an algebraic number field, L be a normal extension of K with Galois group G. Let C be a conjugacy class of G, and define

πC (x; L/K) = #{p : p is unramiﬁed in L, [(L/K)/p] = C,NK/Q(p) ≤ x} where the [(L/K)/p] denotes the conjugacy class of Frobenius automorphisms corresponding to prime ideals P ⊃ p. Finally, dL will denote the absolute value of the discriminant of L, and nL = [L : Q]. CHAPTER 1. INTRODUCTION 14

The following theorem of Lagarias and Odlyzko [39] will be used throughout:

Theorem 1.3.1 (Eﬀective Chebotarev Density Theorem). We have

#C #C − 1 1 β0 2 πC (x; L/K) − li(x) ≤ li(x ) + O x exp −c1n (log x) 2 #G #G L

where β0 is the exceptional real zero of ζL(s), assuming it exists. If it does not exist

this term is omitted from the above. Also, β0 can be eﬀectively computed in terms of

dL and nL. Assuming the GRH for the Dedekind-zeta function of L, we have #C #C √ π (x; L/K) = li(x) + O x (log(d xnL ) + log d ) . C #G #G L L All constants are eﬀectively computable. The implied constants are absolute.

A non-eﬀective version of this theorem was ﬁrst proved by Chebotarev [8, 9]. Improvements in the error term have been made by Serre [62], Murty, Murty and Saradha, [52], and Murty and Murty [51].

1.4 Organization of the Thesis

In Chapter 2, we introduce a new technique that allows to solve problems related to the summation X f(ia(p)). p≤x In particular, we look at the functions f(n) = d(n) and f(n) = (log n)α where 0 < α < 1 is ﬁxed. In Chapter 3, we generalize the problem of determining X f(ia(p)) ∼ Af (a)li(x) p≤x CHAPTER 1. INTRODUCTION 15

∗ as x → ∞ to the r-rank case. That is, let Γ = ha1, a2, . . . , ari ⊂ Q where a1, a2, . . . , ar is a set of multiplicatively independent integers. Then, we can reduce Γ modulo p and consider it as a subgroup of (Z/pZ)∗. Thus, we can consider the above problem. We obtain asymptotic formulæ for f(x) = log(x) and for f(x) = w where w ∈ N is ﬁxed assuming the GRH. In Chapter 4, we generalize the above problem to elliptic curves. That is, let

E be an elliptic curve defined over Q. Then, E(Q) is a finitely generated Abelian group by the Mordell-Weil theorem. As above, we can let Γ = hP1,P2,...,Pri be a subgroup of E(Q). We can reduce both E(Q) and Γ modulo p and this relation will still hold. Thus, we can consider this problem in the elliptic curve setting. We obtain an asymptotic formula for f(n) = ω(n) and upper and lower bounds for f(n) = Ω(n) and log n assuming the GRH. In Chapter 5, we consider the above problem in classical case and prove asymptotic formulæ for f(n) = ω(n) and Ω(n). We also prove that for f(x) = log x, the average summation, and its second moment are as the conjecture predicts. We also revisit a result from the author’s Master’s project [19] and obtain a result related to the index and a second conjecture mentioned in Fomenko [21]. In Chapter 6, we investigate the above problems in relation to a conjecture of Erd˝os.In particular, we show that Conjecture A of Hooley disappears if we assume this conjecture and a quasi-GRH. In Chapter 7, a related conjecture is also heuristically analysed. Also, we discuss the difficulties of obtaining an asymptotic formulæ for f(x) = log x in the classical setting, and f(n) = Ω(n) and f(x) = log x in the elliptic curve setting. Finally, we discuss future research. Chapter 2

A New Method

2.1 Introduction

As we saw in §1.2, we can ask when does the following relation hold for f : N → C: X f(ia(p)) ∼ caπ(x), p≤x where ca is a constant dependent on at most a and f? We note that if we write X f(n) = g(d), d|n where g : N → C, then X X X X X f(ia(p)) = g(d) = g(d) 1

p≤x p≤x d|ia(p) d≤x p≤x d|ia(p) X = g(d)πd(x), d≤x

16 CHAPTER 2. A NEW METHOD 17

where πd(x) := #{p ≤ x : d|ia(p)}. We note that it is always possible to write X f(n) = g(d) d|n by the M¨obiusInversion Formula (see [12, §1.2]).

Assuming g : N → C is well-behaved, we have no difficulty applying standard techniques to obtain results of this nature. For example, f(n) = ω(n) and f(n) = Ω(n) fall into this category (see §5.2). However, more complicated functions cause difficulties if we try to use the standard techniques. For example, f(n) = log n or f(n) = d(n) cause difficulties when considering intermediate primes and large divisors, respectively. To see this, let us consider the following summations X X X log(ia(p)) = Λ(d)πd(x) + Λ(d)πd(x) √ √ √ p≤x d≤ x x

where πd(x) := #{p ≤ x : d|ia(p)} and A, B > 0 are arbitrarily ﬁxed. (In general, when we write a generic sum over primes as X X X X = + + p≤x p≤y y

summation adequately without assuming something beyond the reach of GRH and the Chebotarev density theorem. This is why Fomenko [21] assumed Conjecture A of Hooley. Similar difficulties exist for f(n) = d(n), but they are more difficult because in the last summation Hooley’s argument [31, Equation (3)] no longer applies. In this chapter, we will discuss a new technique that eliminates some of these difficulties if we only assume GRH. For example, we will prove the following theorem in §2.3:

Theorem 2.1.1. Suppose a ∈ Z is not 0 or ±1. Let α ∈ (0, 1) be ﬁxed. Suppose fur-

1/n ther that the GRH holds for the Dedekind zeta functions of the ﬁelds Kn = Q(ζn, a )

th where ζn is a primitive n root of unity. Then, for any ε > 0, we have X x (log i (p))α = c li(x) + O , a a (log x)2−ε−α p≤x

where ca is a constant.

In §2.4, we will prove the following theorem:

Theorem 2.1.2. Suppose a ∈ Z is not 0 or ±1. Suppose further that the GRH holds

1/n for the Dedekind zeta functions of the ﬁelds Kn = Q(ζn, a ) where ζn is a primitive nth root of unity. Then, for any ε > 0, we have X x d(i (p)) = c0 li(x) + O , a a (log x)2−ε p≤x where

0 X 1 ca = , [Kd : ] d≥1 Q a constant.

Before we can prove these theorems, we need some preliminary lemmata. CHAPTER 2. A NEW METHOD 19

2.2 Lemmata

Recall the following theorem of Pappalardi [54]:

Theorem 2.2.1 (Pappalardi). Suppose a ∈ Z is not 0 or ±1. Suppose further that

1/n the GRH holds for the Dedekind zeta functions of the ﬁelds Kn = Q(ζn, a ) where

th ζn is a primitive n root of unity. Then, we have X x log log x log(i (p)) . a log x p≤x From this, we immediately have

Corollary 2.2.1. Suppose a ∈ Z is not 0 or ±1. Suppose further that the GRH holds

1/n for the Dedekind zeta functions of the ﬁelds Kn = Q(ζn, a ) where ζn is a primitive nth root of unity. Then, we have x log log x #{p ≤ x : i (p) > y} . a (log x)(log y)

To see this note that X x log log x log y#{p ≤ x : i (p) > y} ≤ log i (p) . a a log x p≤x Dividing both sides by log y gives us the corollary. We call the following statement the quasi-Riemann hypothesis: For K an

algebraic number ﬁeld with Dedekind-zeta function ζK (s), there exists ε ∈ (0, 1/2) such that if <(s) > 1 − ε, then ζK (s) 6= 0. We note that in this case, the eﬀective Chebotarev Density Theorem (mimic [39, Theorem 1.1] and [62, Theorem 4]) becomes     

#C  1−ε   Y   πC (x; L/K) = li(x) + O #Cx #G  p x , #G   p ramiﬁes   in K/Q where all of the above symbols are as in §1.3 and the O-constants depend only on ε. CHAPTER 2. A NEW METHOD 20

Let πd(x) := #{p ≤ x : d|ia(p)}. We claim that with the quasi-Riemann hypothesis we have

li(x) 1−ε πd(x) = 1/d + O x log(adx) [Q(ζd, a ): Q] th where ζd is a primitive d -root of unity. To see this we note that d|ia(p) if and

p−1 only if p ≡ 1 mod d and a d ≡ 1 (mod p). The second condition is equivalent to xd ≡ a mod p has a solution modulo p. This along with the ﬁrst condition says

d the polynomial x − a splits into d distinct linear factors over Fp[x]. Thus, p splits

1/d completely in Q(ζd, a ) (see [18, Theorem 5.5.1]). Also, if d - ia(p), then p 6≡ 1 mod d p−1 or p ≡ 1 (mod d) and a d 6≡ 1 (mod p). If p 6≡ 1 mod d, then p does not split

completely in Q(ζd) (see [53, Corollary 10.4 of Chapter 1]) and hence, does not split

1/d completely in Q(ζd, a ) (see [37, Property 2.3 of Chapter 3]). If p ≡ 1 mod d and

p−1 d d a d 6≡ 1 (mod p), then x 6≡ a mod p. Thus, x − a does not split into d distinct

1/d linear factors in Fp[x]. Hence, p does not split completely in Q(ζd, a ). Thus, d|ia(p)

1/d if and only if p splits completely in Q(ζd, a ). However, this is true if and only if the

1/d 1/d [(Q(ζd, a )/Q), p], the conjugacy class of the Frobenius automorphism of Q(ζd, a ) over Q, must equal 1. Hence, the claim holds upon noting that the discriminant of

1/d d−1 d ϕ(d) Q(a ) is ±a d (see [37, Page 40]), the discriminant of Q(ζd) divides d . Thus, by [37, Theorem 7.3 of Chapter 1], we have Y Y p = p ≤ ad p ramiﬁes p|ad in K/Q 1/d 1/d 2 Also, #G = [Q(ζd, a ): Q] ≤ [Q(ζd): Q][Q(a ): Q] ≤ dϕ(d) ≤ d . Thus,

li(x) 1−ε πd(x) = 1/d + O x log(adx) [Q(ζd, a ): Q] We will ignore the a in the error term as it is a ﬁxed throughout this chapter. That CHAPTER 2. A NEW METHOD 21

is,

li(x) 1−ε πd(x) = 1/d + O x log(dx) [Q(ζd, a ): Q] We also have the following lemma:

Lemma 2.2.1. Let r and k be ﬁxed positive integers. Suppose the quasi-Riemann

1/n th hypothesis for the ﬁelds Kn = Q(a , ζn) where ζn is a primitive n root of unity. Then,

X r dk(ia(p)) π(x), p≤x where the implied constant is dependent on at most r and k.

Proof. [24, Corollary 22.11] states that for r and k positive integers, we have

r(k−1)t log t r X dk(n) ≤ (2d(m)) log 2 . m|n m≤n1/t The general idea of truncating the range of interest in the above summation originates with work van der Corput [70]. This above development was ﬁrst initiated by Lan- dreau [40] and continued by Iwaniec and Munshi [35], and Friedlander and Iwaniec [23].

r(k−1)t log t Let A = log 2 . We will see that our choice of t is bounded and thus A is bounded. This will be important in the sequel. Let us recall some facts about d(m): A X d(m) [A]+1 (log x)2 m n≤x and d(m) mδ for any δ > 0 (see [12, Exercises 10.5.3 and 1.5.3, respectively]). CHAPTER 2. A NEW METHOD 22

Thus,

X r X X A X A dk(ia(p)) ≤ (2d(m)) d(m) πm(x)

p≤x p≤x m|ia(p) m≤x1/t m≤x1/t   X d(m)A X li(x) + O x1−ε log x d(m)A mϕ(m)   m≤x1/t m 0, and this last term can be seen ([12, Exercise 5.5.3]) to be bounded for any δ < 1. Therefore, the result holds.

Note that the above results give the following lemma:

Lemma 2.2.2. Suppose a ∈ Z is an integer diﬀerent from 0 or ±1. Suppose further

1/n that the GRH holds for the Dedekind zeta functions of the ﬁelds Kn = Q(ζn, a )

th where ζn is a primitive n root of unity. Then, we have X x d (i (p))r k a (log x)β p≤x√ x ia(p)> (log x)B for any β < 2 and B ∈ R ﬁxed.

1 1 Proof. To see this, let p, q > 1 be real numbers such that p + q = 1. Then, by H¨older’s CHAPTER 2. A NEW METHOD 23

inequality and Corollary 2.2.1, we have 1 1   p   q     X r  X   X rq dk(ia(p)) ≤  1  dk(ia(p))      p≤x p≤x p≤x √  √   √  x x x ia(p)> ia(p)> ia(p)> (log x)B (log x)B (log x)B 1 1 π(x) log log x p 1 π(x)(log log x) p q (π(x)) 1 log x (log x) p 1 x(log log x) p . 1+ 1 (log x) p The result now follows upon noting that we may choose p > 1 arbitrarily close to 1 and that log log x (log x)δ for every δ > 0.

2.3 The Function (log n)α

In this section, we are going to prove Theorem 2.1.1. Let α < 1 be a ﬁxed real number. Write X (log n)α = g(d). d|n Then, by the M¨obiusInversion Formula, we have X nα g(n) = µ(d) log , d d|n and so, X |g(n)| ≤ (log n)α 1 = (log n)αd(n). d|n CHAPTER 2. A NEW METHOD 24

We have

X α X (log(ia(p))) = g(m)πm(x) p≤x m≤x X X = g(m)πm(x) + g(m)πm(x), m≤y y 0, we have X g(m) log y = c + O mϕ(m) a y1−δ m≤y

for any δ > 0, and where ca is a constant. Thus, X x g(d)πd(x) = cali(x) + O B−2−α . √ (log x) d≤ x (log x)B CHAPTER 2. A NEW METHOD 25

Also, we have

X X X α X X g(m)πm(x) = g(m) ≤ (log x) d(m)

yy m|ia(p) m|ia(p) X X ≤ (log x)α d(m)

p≤x m|ia(p) ia(p)>y

α X 2 ≤ (log x) d3(ia(p)) p≤x ia(p)>y x , (log x)β−α by Lemma 2.2.2, from the fact X dk+1(n) = dk(n) d|n √ x for all positive integers n and k, and since y = (log x)B for some fixed B ∈ R. Therefore, Theorem 2.1.1 holds by letting B > 3+α since β can arbitrarily close to 2 and α < 1. We should note that this technique will not work for α = 1 unless we can improve upon results of the following flavour π(x) log log x #{p ≤ x : i (p) > y} , a log y or prove related results about the number of primes with divisors in a specified range. The work of Erd˝osand Murty [17], and continuation by Ford [22] have shown that it is possible to obtain non-trivial upper bounds for √ x √ # p ≤ x : p − 1 has a divisor in , x(log x)B . (log x)B However, these bounds do not resolve the problem for α = 1. CHAPTER 2. A NEW METHOD 26

2.4 The Divisor Function

In this section, we will prove Theorem 2.1.2. Recall the following theorem of Fomenko [21]:

Theorem 2.4.1 (Fomenko). Let a be a squarefree integer not 0 or ±1. Let w ∈ N

be ﬁxed. Suppose the GRH holds for the Dedekind-zeta functions of the ﬁelds Kn =

1/n th Q(a , ζn) where ζn is a primitive n root of unity. Deﬁne

Na(x; w) := #{p ≤ x : ia(p) = w}.

Then, x log log x x N (x; w) = A (w)li(x) + O + O , a a ϕ(w)(log x)2 (log x)A where X µ(k) Aa(w) = , [Kkw : ] k≥1 Q A is any ﬁxed real number, and the implied constants depend on at most a and A.

We note that this theorem can be extended to a ∈ Z diﬀerent from 0 and ±1 (see [72]). Also, as mentioned before, the above theorem had the conditions w ≤ log x and only the ﬁrst error term. However, the technique shows the above theorem to also be true. We will also need the following fact: X d(n) = 2 1 − δ(n),

d|√n d≤ n where   1 if n is a square δ(n) := .  0 otherwise CHAPTER 2. A NEW METHOD 27

Thus, we have     X X  X  d(ia(p)) = 2 1 − δ(ia(p))   p≤x p≤x d√|ia(p) d≤ ia(p) X X X X X = 2 1 − 2 1 − δ(ia(p)) √ √ d≤ x p≤x d≤ x p≤x p≤x d|ia(p) d|ia(p) 2 ia(p)

2.4.1 The First Summation

We have X X X πd(x) = πd(x) + πd(x), √ √ d≤ x d≤y yy d≤y x log y √ = c li(x) + O + O(y x log x), 1 y log x where X 1 c1 = [Kd : ] d≥1 Q CHAPTER 2. A NEW METHOD 28

√ x is a constant since [Kd : Q] dϕ(d). Choosing y = (log x)B for any ﬁxed real number B gives us X x πd(x) = c1li(x) + O B−1 . √ (log x) d≤ x (log x)B Now, X X X X X πd(x) = 1 = 1 √ √ √ x √ x √ p≤x p≤x x √ B

p≤x√ d|ia(p) p≤x√ x x ia(p)> ia(p)> (log x)B (log x)B x (log x)β for any β < 2 by Lemma 2.2.2. Thus, X x πd(x) = c1li(x) + O β . √ (log x) d≤ x

2.4.2 The Second Summation

Let C be a ﬁxed positive real number. For the second summation, we have d−1 d−1 d−1 X X X X X X Na(x; md) = Na(x; md) + Na(x; md) √ √ C d≤ x m=1 d≤(log x) m=1 (log x)C

By Theorem 2.4.1, we have d−1 d−1 X X X X x log log x x N (x; md) = A (md)li(x) + O + O a a ϕ(md)(log x)2 (log x)A d≤(log x)C m=1 d≤(log x)C m=1

d−1  d−1  X X x log log x X X 1 = li(x) A (md) + a  (log x)2 ϕ(md) d≤(log x)C m=1 d≤(log x)A m=1 x + . (log x)A−2C Now, d−1 X X 1 X log d ϕ(md) ϕ(d) d≤(log x)C m=1 d≤(log x)C

(log log x)2.

x(log log x)3 So, the ﬁrst error term is (log x)2 . It can be shown that 1 A (w) a wϕ(w) for all w ∈ N using [72, Proposition 4.1]. With this, it can be shown that d−1 X X (log log x)2 A (md) = c + O , a 2 (log x)C d≤(log x)C m=1 where X X c2 := Aa(mk) k≥1 1≤m

For the other summations, we have d−1 X Na(md) ≤ πd(x). m=1 Thus, by the Chebotarev density theorem, we have d−1 X X X Na(x; md) ≤ πd(x) √ √ (log x)C

2.4.3 The Third Summation

For the last summation, we have

X X 2 δ(ia(p)) = #{p ≤ x : ia(p) = m } √ p≤x m≤ x

X 2 = Na(x; m ). √ m≤ x CHAPTER 2. A NEW METHOD 31

Now,

X 2 X 2 X 2 Na(x; m ) = Na(x; m ) + Na(x; m ), √ √ m≤ x m≤y y

X 2 X X µ(w) c3 := Aa(m ) = . [K 2 : ] m≥1 m≥1 w≥1 k w Q Thus, X x log log x N (x; m2) = c li(x) + O . a 3 (log x)2 m≤(log x)C For the second summation, we have

X 2 X 2 Na(x; m ) ≤ π(x; m , 1) √ √ (log x)C

for any ﬁxed δ ∈ (0, 1/2). Thus, by the Brun-Titchmarsh Theorem(see [12, Theorem 7.3.1]), we have X X x π(x; m2, 1) ϕ(m2) log x 1 C (log x)C (log x) x X 1 log x m2−ε m>(log x)C x (log x)1+C(1−ε) for any ε > 0. Thus, for C = 1 − ε, we have X x π(x; m2, 1) . (log x)2 1 (log x)C x 2 −δ

1 +δ x 2 .

Thus, X x log log x δ(i (p)) = c li(x) + O . a 3 (log x)2 p≤x Therefore, X x d(i (p)) = (2c − 2c − c )li(x) + O . a 1 2 3 (log x)β p≤x

2.4.4 Computation of the Constant

To ﬁnish the proof, we need to prove X 1 = 2c − 2c − c . [K : ] 1 2 3 n≥1 n Q CHAPTER 2. A NEW METHOD 33

Note that X 1 c = . 1 [K : ] n≥1 n Q

Therefore, we need to prove that c1 = 2c2 + c3. That is, we need to prove

X 1 X X X 2 = 2 Aa(mk) + Aa(k ) [Kn : ] n≥1 Q k≥1 1≤m

2 2 if and only if d = d1d2 with gcd(d1, d2) = 1 such that d1|m and d2|n. Thus, g is multiplicative. Hence,   Y X α−2d g(n) =  µ(p ) pαkn p2d|pα Y = (−1)α = (−1)Ω(n). pαkn Let X X f(n) = 1 = 1. n=mk d|√n m

n This is true since n = mk with m < k if and only if k|n and k < k, and this is true √ if and only if k|n and n < k. However, X X d(n) = 1 = 2 1 + δ(n),

d|n d|√n d< n where   √ 1 if n ∈ Z δ(n) = .  0 otherwise Therefore, d(n) − δ(n) f(n) = . 2 Hence, we will be ﬁnished if we can prove X d(n/w) − δ(n/w) 1 = 2 µ(w) + (−1)Ω(n), 2 w|n or equivalently,   X d(n/w) − δ(n/w) 0 if Ω(n) is even µ(w) = . 2  w|n 1 if Ω(n) is odd By the M¨obiusInversion Formula, all we need to show is that d(n) − δ(n) = #{d|n : Ω(n) is odd}. 2 In order to do this, we need the following lemma about generating functions:

Lemma 2.4.1. Let α1, α2, . . . , αr be positive integers. For all i ∈ {1, 2, . . . , r}, let

2 αi fi(x) = 1 + x + x + ... + x .

Let

X k f(x) = f1(x)f2(x) ··· fr(x) = akx , k≥0 CHAPTER 2. A NEW METHOD 35

where ak = 0 for all k > α1 + α2 + ... + αr.

(a) If there exists i ∈ {1, 2, . . . , r} such that αi is odd, then X X a2k = a2k+1. k≥0 k≥0 That is, the sum of the coeﬃcients to an odd power is equal to the sum of the coeﬃcients to an even power.

(b) If all of α1, α2, . . . , αr are even, then X X a2k = 1 + a2k+1. k≥0 k≥0 That is, the sum of the coeﬃcients to an odd power is equal to 1 plus the sum of the coeﬃcients to an even power.

Proof. (a) We will prove this by induction on r. The result holds for r = 1. Suppose

it is true for r − 1 with r ≥ 2. Let f1(x), f2(x), . . . , fr(x) be as in (a). Without

loss of generality, we may assume that α1 is odd. Therefore, the result holds

∞ k for g(x) = f1(x)f2(x) ··· fr−1(x). Let {bk}k=0 be the coeﬃcients of x for g(x). Then, we have X X b2k = b2k+1. k≥0 k≥0 CHAPTER 2. A NEW METHOD 36

Now,

f(x) = g(x)fr(x) ! X k 2 αr = bkx (1 + x + x + ··· + x ) k≥0

! αr ! X k X m = bkx x k≥0 m=0   ! X 2k X 2k+1  X m X m = b2kx + b2k+1x  x + x  k≥0 k≥0 0≤m≤αr 0≤m≤αr m is even m is odd

X X 2k+m X X 2k+1+m = b2kx + b2k+1x

0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd

X X 2k+m X X 2k+1+m + b2kx + b2k+1x

0≤m≤αr k≥0 0≤m≤αr k≥0 m is odd m is even Therefore, X X X X X a2k = b2k + b2k+1

k≥0 0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd X X X X = b2k+1 + b2k

0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd X = a2k+1. k≥0 Hence, part (a) holds.

(b) Again, we will prove this by induction on r. The result holds for r = 1. Suppose

it is true for r − 1 where r ≥ 2. Let f1(x), f2(x), . . . , fr(x) be as in (b). Let

∞ k g(x) = f1(x)f2(x) ··· fr−1(x). Let {bk}k=0 be the coeﬃcients of x for g(x). Then, we have X X b2k = 1 + b2k+1. k≥0 k≥0 CHAPTER 2. A NEW METHOD 37

Now,

f(x) = g(x)fr(x) ! X k 2 αr = bkx (1 + x + x + ··· + x ) k≥0

! αr ! X k X m = bkx x k≥0 m=0   ! X 2k X 2k+1  X m X m = b2kx + b2k+1x  x + x  k≥0 k≥0 0≤m≤αr 0≤m≤αr m is even m is odd

X X 2k+m X X 2k+1+m = b2kx + b2k+1x

0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd

X X 2k+m X X 2k+1+m + b2kx + b2k+1x

0≤m≤αr k≥0 0≤m≤αr k≥0 m is odd m is even Therefore, X X X X X a2k = b2k + b2k+1

k≥0 0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd ! ! X X X X = 1 + b2k+1 + −1 + b2k

0≤m≤αr k≥0 0≤m≤αr k≥0 m is even m is odd X X X = a2k+1 + 1 − 1

k≥0 0≤m≤αr 0≤m≤αr m is even m is odd X = 1 + a2k+1 k≥0

since αr is even. Hence, part (b) holds.

We claim that Lemma 2.4.1 implies d(n) − δ(n) = #{d|n : Ω(n) is odd}. 2 CHAPTER 2. A NEW METHOD 38

α1 α2 αr Let n = p1 p2 ··· pr be the unique prime power factorization of n. Then, we have r X Y 2 αi d = (1 + pi + pi + ··· + pi ) d|n i=1

X β1 β2 βr = p1 p2 ··· pr .

0≤βi≤αi for all i Thus, X #{d|n : Ω(d) is odd} = 1.

0≤βi≤αi for all i β1+β2+···+βr is odd

2 αi For i ∈ {1, 2, . . . , r}, let fi(x) = 1 + x + x + ··· + x . Let

X k f(x) = f1(x)f2(x) ··· fr(x) = akx . k≥0 Also note that X f(x) = xβ1+β2+···+βr .

0≤βi≤αi for all i Thus, X X a2k+1 = 1.

k≥0 0≤βi≤αi for all i β1+β2+···+βr is odd Similarly, replacing “odd” with “even” gives us X X a2k = 1.

k≥0 0≤βi≤αi for all i β1+β2+···+βr is even Therefore, by Lemma 2.4.1, we have

#{d|n : Ω(d) is odd} = #{d|n : Ω(d) is even} if some αi is odd, and

#{d|n : Ω(d) is odd} + 1 = #{d|n : Ω(d) is even} if all αi are even. This last condition is equivalent to n being a square. Therefore, we CHAPTER 2. A NEW METHOD 39

have

#{d|n : Ω(d) is odd} + δ(n) = #{d|n : Ω(d) is even}.

Hence,

d(n) = #{d|n : Ω(d) is odd} + #{d|n : Ω(d) is even}

= 2#{d|n : Ω(d) is odd} + δ(n).

Thus, d(n) − δ(n) #{d|n : Ω(d) is odd} = , 2 as required. Therefore, Theorem 2.1.2 holds. Chapter 3

Finitely-Generated Analogues

3.1 Statement of Theorems

Throughout this chapter, we let {a1, a2, . . . , ar} be a multiplicatively independent set

∗ of rational numbers. Let Γ = ha1, a2, . . . , ari be a subgroup of Q . For every prime number p with νp(a) = 0 for all a ∈ Γ, we deﬁne

Γp := Γ mod p = {a mod p : a ∈ Γ}.

Ai For each i ∈ {1, 2, . . . , r}, write ai = with Ai,Bi ∈ and gcd(Ai,Bi) = 1 and Bi Z

Bi > 0. There is a unique choice for these Ai’s and Bi’s. We note that

{p : νp(a) 6= 0 for some a ∈ Γ} = {p : p|AiBi for some i ∈ {1, 2, . . . , r}}.

Thus, #{p : νp(a) 6= 0 for all a ∈ Γ} < ∞, and so we will ignore this set throughout

∗ this chapter. Notice that Γp is a subgroup of Fp. So, we deﬁne the index of Γ

modulo p as the index of Γp over Fp and denote it by iΓ(p). That is,

∗ p − 1 iΓ(p) := [Fp :Γp] = . |Γp|

40 CHAPTER 3. FINITELY-GENERATED ANALOGUES 41

We also deﬁne fΓ(p) := |Γp|.

3.1.1 The r-rank Theorem

We are interested in computing X log(iΓ(p)). p≤x If r = 1, then the conjecture is X log(ia(p)) = c(a)li(x) + o(li(x)) p≤x where X Λ(n) c(a) = [K : ] n≥1 n Q √ n th where Kn = Q( a, ζn) with ζn a primitive n -root of unity and a ∈ Z is not 0 or ±1. It should be noted that this conjecture is resolved assuming the GRH and a conjecture of Hooley [32]. Conjecture A of Hooley is the following:

Conjecture (Conjecture A). Let Pb(y; `, t) be the number of primes p ≤ y such that

t th 1 2 b is an ` -power residue modulo p and for which `|p − 1. Then, for y 4 < ` < y, we have y P (y; `, t) b ϕ(`)(log(2y/`))2 where the implied constant is absolute.

Theorem 3.1.1 (Fomenko [21]). Suppose a is a squarefree integer. Assume the GRH

1/n th holds the Dedekind zeta functions of Q(ζn, a ) where ζn is a primitive n root of CHAPTER 3. FINITELY-GENERATED ANALOGUES 42

unity, and that Conjecture A of Hooley holds. Then, we have X X Λ(n) x log log x log(i (p)) = li(x) + O . a [K : ] (log x)2 p≤x n≥1 n Q

√ y √ 2i In fact, letting t = 0 and restricting ` to the range (log y)4 , y(log y) is all that is needed to prove the above theorem since this gives us y P (y; `, 0) b ϕ(`)(log y)2 in this range. However, one should immediately note that this is not only a difficult conjecture to prove, it is also a difficult conjecture to disprove. If it is theoretically provable, one needs to find the correct absolute constant. However, we could potentially continue to enlarge the constant never knowing when to stop. So, this conjecture is difficult to disprove. In fact, one would need to show that P (y; `, t) lim sup b = ∞. y→∞ y/ϕ(`)(log(2y/`))2 y1/4<`≤y 1/4 That is, for every y, we need to find the `y ∈ (y , y) that corresponds to the maximum value for

Pb(y; `, t) y/ϕ(`)(log(2y/`))2 1/4 with ` ∈ (y , y). Finding this `y will become more difficult as y → ∞. Also, calculating Pb(y; `, t) is highly non-trivial. We also note that in the above range it is sufficient to assume the Pair Correlation Conjecture instead of Conjecture A of Hooley. For a formulation of this conjecture, see Murty and Murty [51]. In fact, this conjecture allows us to obtain error terms which are significantly better than in Theorem 1.1.1 as well as in the above theorem. It also allows us to solve Hooley’s Conjecture A in the above range as long as t is still CHAPTER 3. FINITELY-GENERATED ANALOGUES 43

equal to 0. In §3.2, we will prove a similar result for r ≥ 2. However, we will not need the higher-rank conjecture A of Hooley to do this.

1/n 1/n 1/n Theorem 3.1.2. Suppose GRH holds for the ﬁelds Q(ζn, a1 , a2 , . . . , ar ) where

th ζn is a primitive n root of unity, and suppose A > 2. Let cΓ be a constant dependent on Γ. For r = 2, we have

X 9 A+ A log(iΓ(p)) = cΓli(x) + O x 10 (log x) r . p≤x For r = 3, we have 6 ! X x 7 log(i (p)) = c li(x) + O . Γ Γ (log x)A−2 p≤x For r > 3, we have

X 5 A+ A log(iΓ(p)) = cΓli(x) + O x 6 (log x) r . p≤x In fact, we will be able to show that in the above theorem, only δ-GRH is necessary

where δ ∈ [1/2, 1). Here, δ-GRH is the following claim: if <(s) > δ, then ζK (s) 6= 0. Let θ deﬁned as follows:  (1 + δ)(r + 1) δ 1 + 2δ  if < r <  2r + 1 1 − δ 1 − δ  θ := θ(δ, r) = .  2 + δ 1 + 2δ  if r ≥  3 1 − δ 1/n 1/n 1/n Theorem 3.1.3. Suppose δ-GRH holds for the ﬁelds Q(ζn, a1 , a2 , . . . , ar ) where

th δ ζn is a primitive n root of unity, and suppose A > 2. For r > 1−δ , we have

X X log q θ A log(iΓ(p)) = li(x) + O x (log x) 2 , n α p≤x qα≥2 q 1/d 1/d 1/d where nd = [Q(ζd, a1 , a2 , . . . , ar ): Q] and the implied constant is dependent only on A, Γ, and r. CHAPTER 3. FINITELY-GENERATED ANALOGUES 44

Both in the above theorem and δ-GRH theorem we obtain power-saving results.

That is, all of our error terms look like xθ(log x)A with θ < 1 and A ∈ R ﬁxed. This is in stark contrast to Theorem 1.1.1, where error terms are of the form x log log x (log x)2 are common and are currently optimal or nearly optimal.

3.1.2 An Immediate Corollary

We have the following corollary of Theorems 3.1.2 and 3.1.3

Corollary 3.1.1. Suppose the hypotheses of Theorem 3.1.2 or Theorem 3.1.3 hold. Then,

X θ A log(fΓ(p)) = x − cΓli(x) + O x (log x) p≤x where A ∈ R and θ corresponds to θ in Theorem 3.1.3 or the exponents of x in the O term in Theorem 3.1.2, and where the implied constant is dependent only on A, Γ, and r.

Proof. Note that X X X log(fΓ(p)) = log(p − 1) − log(iΓ(p)) p≤x p≤x p≤x X X p − 1 X = log p + log − log(i (p)) p Γ p≤x p≤x p≤x X X = log p − log iΓ(p) + O(log log x). p≤x p≤x We have used the Taylor series for log(1 − x) with |x| < 1 to obtain X p − 1 log log log x. p p≤x CHAPTER 3. FINITELY-GENERATED ANALOGUES 45

Also, from Theorem 3.1.2 or 3.1.3, we have

X θ A log(iΓ(p)) = cΓli(x) + O(x (log x) ) p≤x where we take A to be the corresponding exponents of the log x terms in Theorems 3.1.2 and 3.1.3, and θ as the corresponding exponents of the x terms in these theorems. So we need to evaluate X log p. p≤x We note that by the Aramata-Brauer Theorem (see [18, Chapter 11, §4] or [50, Chapter 2, §3]), the δ-Riemann hypothesis is also true since our extension is Galois. That is, there are no zeros of the Riemann zeta function in the region <(s) > δ. Therefore, we may assume

π(x) = li(x) + O(xδ log x), or equivalently, X θ(x) := log p p≤x

= x + O(xδ(log x)2).

Putting all of this together we obtain

X δ 2 θ A log(fΓ(p)) = x + O(x (log x) ) − cΓli(x) + O(x (log x) ) p≤x

θ A = x − cΓli(x) + O(x (log x) )

because θ > δ. Hence, the corollary is true. CHAPTER 3. FINITELY-GENERATED ANALOGUES 46

3.1.3 The Fixed Index Theorem

Let w ∈ N be fixed, and let a be an integer different from 0 and ±1. Define Na(x; w)

be the number of primes p ≤ x such that ia(p) = w. Fomenko [21] was also able to show (following Stephens [67]) the following theorem:

1/n Theorem 3.1.4 (Fomenko). Suppose GRH holds for the ﬁelds Kn := Q(ζn, a )

th where ζn is a primitive n root of unity. Then for any A > 2, we have x log log x x N (x; w) = A (w)li(x) + O + O a a ϕ(w)(log x)2 (log x)A where ∞ X µ(k) Aa(w) := [Kkw : ] k=1 Q is a constant, and the implied constants are dependent only on a and A.

It should be noted that in Fomenko’s original theorem w was ≤ log x, a > 1 was

x squarefree, and the error term (log x)A was missing. However, the proof technique allows for the above stronger statement. In §3.3, we will generalize the above result to rank r ≥ 2. That is, we will prove the following theorem:

Theorem 3.1.5. Let δ < r/(r + 1). Suppose δ-GRH holds for the Dedekind zeta

1/n 1/n 1/n th functions for the ﬁelds Q(ζn, a1 , a2 , . . . , ar ) where ζn is a primitive n root of 1 r unity. Let α > r + 1, and let θ = α + r+1 < 1. Then, x N (x; w) = δ li(x) + O + O(xθ log x) Γ Γ,w ϕ(w)w(log x)2 where X µ(m) δ = Γ,w [ (ζ , Γ1/mw): ] m≥1 Q mw Q CHAPTER 3. FINITELY-GENERATED ANALOGUES 47

is a constant.

3.2 An Asymptotic Formula

For d ∈ N, deﬁne

πd(x) := #{p ≤ x : d|iΓ(p)}.

Then, we have X X log(iΓ(p)) = Λ(d)πd(x). p≤x d≤x So, in order to prove this result it suﬃces to prove it for the last summation. We note that by the Lagarias and Odlyzko [39] version of the Chebotarev Density Theorem we have

li(x) √ r+2 πd(x) = + O x log(xd P ) [Kd : Q] 1/d where P is the product of primes p with νp(a) 6= 0 for some a ∈ Γ, Kd = Q(ζd, Γ )

1/d 1/d and Γ := {a : a ∈ Γ}. Let nd = [Kd : Q]. Thus, for some y to be chosen later, we have X X li(x) √ α(r+2) Λ(d)πd(x) = log q + O x log(xq P ) nqα d≤y qα≤y ! X log q X √ = li(x) + O log q x (log x + α(r + 2) log q + log P ) . n α qα≤y q qα≤y Now, X log q X log q X log q = − . n α n α n α qα≤y q qα≥2 q qα>y q CHAPTER 3. FINITELY-GENERATED ANALOGUES 48

1/d We note that the ﬁrst summation above is a constant since nd ≥ [Q(ζd, a1 ): Q] dϕ(d). For the second summation, we use [12, Exercise 5.5.3]. We have X log q X log q 1 α αr . n α ϕ(q )q y qα>y q qα>y Thus, X X log q li(x) Λ(d)πd(x) = li(x) + O nqα y d≤y qα≥2 ! X √ + O log q x (log x + α(r + 2) log q + log P ) qα≤y Now, √ X √ X √ x log x log q = x log x Λ(n) y x log x. qα≤y n≤y Also, √ X √ X √ x α(r + 2)(log q)2 (r + 2) x log y log n y x(log y)2, qα≤y n≤y and X X log P log q = log P Λ(n) y log P. qα≤y n≤y All of these follow from analytic number theory. Therefore, we have X X log q li(x) √ √ 2 Λ(d)πd(x) = li(x) + O + y x log x + y x(log y) . nqα y d≤y qα≥2 Now we must deal with X Λ(d)πd(x). y

Lemma 3.2.1 (Gupta and Murty). Suppose r ≥ 2. Then

1+ 1 #{p : |Γp| ≤ t} t r where the implied constant is dependent is on at most r and a1, a2, . . . , ar.

Then, we have X X X X Λ(d)πd(x) = (log q)πqα (x) ≤ log x 1 y

X α = log x #{p ≤ x : q |iΓ(p)} y

X α X X α #{p ≤ x : q |iΓ(p)} = #{p ≤ x : q|iΓ(p)} + #{p ≤ x : q |iΓ(p)}. y1 Let √ x y = . (log x)4 We claim

X α 3 2 #{p ≤ x : q |iΓ(p)} x 4 (log x) . y1 CHAPTER 3. FINITELY-GENERATED ANALOGUES 50

To see this we note that

X α X x x 3 2 #{p ≤ x : q |i (p)} √ = x 4 (log x) . Γ qα y y1 α>1 We also claim X x #{p ≤ x : q|i (p)} # p : |Γ | ≤ . Γ p y y

once on the left-hand side. That is, there may exist distinct primes q1, q2, . . . , qn ∈

(y, x] such that qi|iΓ(p). Since q1, q2, . . . , qn ∈ (y, x] are distinct, n x 2 = yn < q q ··· q < i (p) < x. (log x)4n 1 2 n Γ Therefore, n ≤ 2. Thus, X x #{p ≤ x : q|i (p)} # p : |Γ | ≤ , Γ p y y

r+1 4+ 4 = x 2r (log x) r . CHAPTER 3. FINITELY-GENERATED ANALOGUES 51

Therefore, we have X X X log(iΓ(p)) = (log q)πqα (x) + (log q)πqα (x) p≤x qα≤y y

3 2 r+1 4+ 4 + O x 4 (log x) + x 2r (log x) r X log q √ x 3 r+1 4 4 4 2 2r 4+ r = li(x) + O x(log x) + 2 + x (log x) + x (log x) n α (log x) qα≥2 q X log q x = li(x) + O 2 . n α (log x) qα≥2 q We should note that y above was not optimally chosen. To see this, let   2  if r = 2  5  α = 5 . 14 if r = 3    1  3 if r ≥ 4 1 We note α ≥ 3 for all r ≥ 2. Let xα y = (log x)A where A > 0 is ﬁxed. Then, all the above arguments are still valid (assuming instead of n ≤ 2 we need CHAPTER 3. FINITELY-GENERATED ANALOGUES 52

to take n ≤ 3). Therefore, r+1 ! r X X log q li(x) √ √ 2 x x log(iΓ(p)) = li(x) + O + y x log x + y x(log y) + √ + n α y y y p≤x qα≥2 q 1 +α ! X log q 1−α A x 2 = li(x) + O x (log x) + O A−1 n α (log x) qα≥2 q 1 +α ! x 2 1− α A (1−α)(r+1) A+ A + O + O x 2 (log x) 2 + O x r (log x) r (log x)A−2

1 +α ! X log q 1−α A x 2 = li(x) + O x (log x) + O A−2 n α (log x) qα≥2 q

1− α A (1−α)(r+1) A+ A + O x 2 (log x) 2 + O x r (log x) r .

1 1 We note that for every choice of α, we have 3 ≤ α < 2 . Hence, all but the last error term above satisfy xΘ with Θ < 1. So, in order to determine if we have any power saving, we only need to verify that (1 − α)(r + 1) < 1. r 1 2 1 This is equivalent to α > 1+r . For r = 2, this is equivalent to 5 > 3 , which is true. 1 For r > 2, this is equivalent to α > 4 , which is true. So, we do indeed have power saving. To ﬁnish we would like to determine 1 α (1 − α)(r + 1) min max 1 − α, + α, 1 − , . α 2 2 r We note that the above is the smallest exponent of x possible as each of those terms correspond to an error term. Analyzing this quantity will yield points of intersections

1 α (1−α)(r+1) related to the lines 1 − α, 2 + α, 1 − 2 , and r where α is a variable and r is ﬁxed. These points of intersection give us the optimal error terms in Theorem 3.1.2. Thus, the result holds. CHAPTER 3. FINITELY-GENERATED ANALOGUES 53

The same can be done for δ-GRH. We note that this implies

li(x) δ r+2 πd(x) = + O x log(xd P ) . [Kd : Q] where P is the same as before. Let  (1 + δ)(r + 1) δ 1 + 2δ  if < r <  2r + 1 1 − δ 1 − δ  θ := θ(δ, r) = .  2 + δ 1 + 2δ  if r ≥  3 1 − δ Since θ > 0, all the arguments from the previous theorem are still true. Thus, we have δ+α X X log q 1−α A x log(iΓ(p)) = li(x) + O x (log x) + O A−2 n α (log x) p≤x qα≥1 q

1− α A (1−α)(r+1) A r+1 + O x 2 (log x) 2 + O x r (log x) r ,

α and since 1 − α ≤ 1 − 2 , we have δ+α X X log q x α A 1− 2 2 log(iΓ(p)) = li(x) + O A−2 + O x (log x) n α (log x) p≤x qα≥1 q

(1−α)(r+1) A r+1 + O x r (log x) r .

Therefore, we need α (1 − α)(r + 1) min max δ + α, 1 − , 2 r α (1−α)(r+1) with respect to α. Note that the terms δ + α, 1 − 2 , and r are the exponents of x in the error terms above. So the above is looking for the smallest possible error terms with respect to the exponent of x. This leads to three α’s of interest: r(1 − δ) + 1 2 2(1 − δ) , , . 2r + 1 r + 2 3 α (1−α)(r+1) These correspond points of intersection between the lines δ + α, 1 − 2 , and r CHAPTER 3. FINITELY-GENERATED ANALOGUES 54

where α is treated as the variable and δ and r are ﬁxed. Substituting these choices of α lead to error terms (1 + δ)(r + 1) 2 + δ , 2r + 1 3 respectively, and these are how θ is deﬁned, and this completes the proof of Theorem 3.1.3 with X log q cΓ = . n α qα≥1 q δ It should be noted that θ < 1 for all δ ∈ [1/2, 1) and r > 1−δ . To see this we note that for case 1, (1 + δ)(r + 1) < 1 2r + 1 if and only if

r + 1 + rδ + δ < 2r + 1

if and only if δ r > , 1 − δ and for case 2, we have 2 + δ 3 < = 1. 3 3 Therefore, as when we assumed GRH, we have obtained power-saving error terms

δ as long as r > 1−δ . We can use the above inequalities on r to determine corresponding inequalities for δ if so desired. Doing this yields the following theorem:

Theorem 3.2.1. Let r ≥ 2 and let δ < r/(r + 1). Suppose δ-GRH holds for the ﬁelds

1/n 1/n 1/n th Q(ζn, a1 , a2 , . . . , ar ) where ζn is a primitive n root of unity. Then,

X X log q θ A log(iΓ(p)) = li(x) + O(x (log x) 2 ), n α p≤x qα≥1 q CHAPTER 3. FINITELY-GENERATED ANALOGUES 55

1/d 1/d 1/d where θ was deﬁned as before and nd := [Q(ζd, a1 , a2 , . . . , ar ): Q].

3.3 The Fixed Index Theorem

For w ∈ N ﬁxed, deﬁne

NΓ(x; w) = #{p ≤ x : iΓ(p) = w}.

We will follow the techniques of Fomenko [21] (who followed Stephens [67]) and Pap- palardi [55] which is essentially Hooley’s technique mutatis mutandis. As such, we deﬁne the following functions:

NΓ(x, y; w) := #{p ≤ x : w|iΓ(p), wq - iΓ(p) for all q ≤ y}

MΓ(x, y, z; w) := #{p ≤ x : wq|iΓ(p) for some q with y ≤ q ≤ z} and

MΓ(x, z; w) := #{p ≤ x : wq|iΓ(p) for some q > z} where y and z are dependent on x and will be chosen later. We note that x − 1 N (x; w) = N x, ; w . Γ Γ w

To see this we note that if p ≤ x contributes to NΓ(x; w), then iΓ(p) = w. This implies w|iΓ(p) and for all primes q, wq - iΓ(p). So, the prime number p contributes to NΓ(x, (x − 1)/w; w). If p ≤ x contributes to NΓ(x, (x − 1)/w; w), then w|iΓ(p) and for all q ≤ (x − 1)/w we have wq - iΓ(p). So, iΓ(p) = wm for some m ∈ N.

If q|m for some prime q, then q|(iΓ(p)/w) ≤ (x − 1)/w, and so q ≤ (x − 1)/w, a contradiction. So m = 1, and the prime number p contributes to NΓ(x; w). Thus, we have NΓ(x; w) = NΓ(x, (x − 1)/w; w). CHAPTER 3. FINITELY-GENERATED ANALOGUES 56

We also have

NΓ(x, y; w) − MΓ(x, y, z; w) − MΓ(x, z; w) ≤ NΓ(x; w) ≤ NΓ(x, y; w) for any y with y ≤ z ≤ (x − 1)/w. Thus, we have

NΓ(x; w) = NΓ(x, y; w) + O(MΓ(x, y, z; w)) + O(MΓ(x, z; w)).

We will evaluate each of these terms separately. For m ∈ N squarefree, let

πm(x) := #{p ≤ x : wq|iΓ(p) for all q|m}.

Then, by a standard inclusion-exclusion argument, we have X0 NΓ(x, y; w) = µ(m)πm(x) m where 0 indicates m runs through the set

{n ∈ N : n is squarefree and p|n implies p ≤ y}.

Since m is squarefree, this implies Y m ≤ q = eθ(y) q≤y where X θ(y) := log p. p≤y

We note that wq|iΓ(p) if and only if p - wqa1a2 ··· ar and p splits completely in

1/wq 1/wq 1/wq Kq := Q(ζwq, a1 , a2 , . . . , ar ).

If we let

1/wm 1/wm 1/wm Km = Q(ζwm, a1 , a2 , . . . , ar ), CHAPTER 3. FINITELY-GENERATED ANALOGUES 57

the compositum of Kq as q ranges over prime divisors of m, then we obtain wm|iΓ(p)

if and only if p - wma1a2 ··· ar and p splits completely in Km. Therefore,

πm(x) = #{p ≤ x : p is unramiﬁed and splits completely in Km}.

By the eﬀective version of the Chebotarev density theorem [39], we have

li(x) √ 1 πm(x) = + O x log x|dm| nm nm

where nm = [Km : Q] and dm is the discriminant of Km. Hensel’s inequality (see [62]) states for a ﬁnite ﬁeld extension K of Q, we have   X log |dK | ≤ nK  log p + log nK 

p|dK

where nK = [K : Q] and dK is the discriminant of K. Therefore,

1/nm Y |dm| ≤ nm p,

νp(ai)6=0 for all i∈{1,2,...,r} However,

r r+1 nm = [Km : Q] ϕ(wm)(mw) ≤ (mw)

1/wm since Km is also the compositum of the ﬁelds Q(ζwm, ai ). Thus, we have X0 NΓ(x, y; w) = µ(m)πm(x) m X0 li(x) √ = µ(m) + O x log(x(mw)r+1P ) n m m ! ! ! X µ(m) X li(x) X0 √ = li(x) + O + O x log x(wm)r+1P . n n m≥1 m m>y m m We have

1/wm nm ≥ [Q(ζwm, a1 ): Q] ϕ(wm)(wm) CHAPTER 3. FINITELY-GENERATED ANALOGUES 58

where the implied constant is dependent only on a1. Thus, we have the following inequalities: X 1 X 1 1 X 1 n ϕ(wm)(wm) ϕ(w)w ϕ(m)m m>y m m>y m>y 1 . ϕ(w)wy Therefore, as long as y → ∞ as x → ∞, we have X µ(m) δ := Γ,w n m≥1 m is absolutely convergent and X li(x) = o(li(x)). n m>y m We must consider X0 √ √ X0 x log(x(wm)r+1P ) = x (log x + (r + 1)(log w + log m) + log P ). m m As long as m ≤ eθ(y) ≤ x and w ≤ x we can bound the above summation by √ X0 √ x log x 1 2π(y) x log x. m This last bound is o(li(x)) if and only if √ x 2π(y) = o . (log x)2 Choosing y = (log x)/6 gives us π(y) ≤ 2y/ log y ≤ (log x)/3. So, 2π(y) ≤ eπ(y) ≤ √ e(log x)/3 = 3 x. Thus, we have li(x) 5 N (x, y; w) = δ li(x) + O + O(x 6 log x). Γ Γ,w ϕ(w)wy

In order to bound MΓ(x, z; w) we recall Lemma 3.2.1 of Gupta and Murty [26] which states

1+ 1 #{p : |Γp| ≤ t} t r . CHAPTER 3. FINITELY-GENERATED ANALOGUES 59

Note that

MΓ(x, z; w) = #{p ≤ x : wq|iΓ(p) for some q > z} n x o = # p ≤ x : |Γ | ≤ p wz n x o ≤ # p : |Γ | ≤ p wz 1+ 1 x r . wz Finally, we have

MΓ(x, y, z; w) = #{p ≤ x : wq|iΓ(p) for some q ∈ [y, z]}

≤ #{p ≤ x : p is unramiﬁed and splits completely in Kq for some q ∈ [y, z]} X ≤ πq(z) y≤q≤z X li(x) √ = + O x log(xqr+1wr+1P ) n y≤q≤z q X 1 √ = li(x) + O(π(z) x(xzr+1wr+1P )) n y≤q≤z q √ li(x) z x log x + O . ϕ(w)wy log z Therefore, we have

NΓ(x; w) = NΓ(x, y; w) + O(MΓ(x, y, z; w)) + O(MΓ(x, z; w)) li(x) = δ li(x) + O + O(x5/6 log x) Γ,w ϕ(w)wy √ 1+ 1 x r z x log x + O + O wz log z x = δ li(x) + O + O(x5/6 log x) Γ,w ϕ(w)w(log x)2 upon choosing y = (log x)/6 and z = x1/(r+1)(log x)r2/(r+1). We note that, as in §3.2, the full GRH is not needed; we can assume that the

r Dedekind zeta functions for the ﬁelds Km have no zeros in the region <(s) > r+1 . In CHAPTER 3. FINITELY-GENERATED ANALOGUES 60

this case, we need to modify the choice of y. Let y be (log x)/α where α > r + 1. The choice for z is the same as before. These choices when substituted into the above proof give us Theorem 3.1.5. We should also note that we never obtain power-saving in the error term as in §3.2. This is due to the fact that y can never grow faster than c log x for some small c, a constant; while in §3.2, y = xα/(log x)A for 0 < α < 1/2 and A > 0. Chapter 4

Elliptic Curve Analogues

Throughout this chapter, we let E be an elliptic curve deﬁned over Q. That is, E : y2 = x3 + Ax + B with A, B ∈ Q. Then it can be shown that E(Q), the solutions to the elliptic curve E with coordinates in Q together with the point at inﬁnity, forms an Abelian group. In fact, the Mordell-Weil Theorem (see [42, 63, 73]) states that

∼ k E(Q) = E(Q)tors × Z ,

where E(Q)tors is finite and corresponds to the points of finite order on E, and k is a non-negative integer. We define k as the rank of E over Q.

This gives us the same setup as in Chapter 3. We had Γ = ha1, a2, . . . , ari ⊂

Q∗. Instead of ﬁnitely generated subgroups of Q∗, we can consider ﬁnitely generated subgroups of E(Q). That is, let P1,P2,...,Pr ∈ E(Q) with r ≤ k, and let

Γ = hP1,P2,...,Pri ⊂ E(Q). In Chapter 3, we set a1, a2, . . . , ar to be multiplicatively independent because we wanted to ignore trivial problems like ha1, a2, . . . , ari =

ha1, a2, . . . , ar, ar+1i where ar+1 is the product of elements a1, a2, . . . , ar. Also, by def- inition of multiplicatively independent, ±1 were always excluded from our possible

61 CHAPTER 4. ELLIPTIC CURVE ANALOGUES 62

choices of a1, a2, . . . , ar because they are points of ﬁnite order. For the same reasons,

we exclude points of ﬁnite order from our choices of P1,P2,...,Pr in the group Γ.

Thus, our multiplicatively independent condition becomes n1P1+n2P2+···+nrPr 6= 0

unless n1 = n2 = ··· = nr = 0. We actually need a stronger condition. Deﬁne

End(E) be the endomorphism ring of E over Q. That is, End(E) is the collection of maps φ : E(Q) → E(Q) such that φ(P + Q) = φ(P ) + φ(Q) for all P,Q ∈ E. Then, E is an End(E)-module by the action of φ · P = φ(P ) for all P ∈ E and

φ ∈ End(E). Then, we say that P1,P2,...,Pr are linearly independent over End(E) if

φ1(P1)+φ2(P2)+···+φr(Pr) = ∞, the point at inﬁnity, then φ1 = φ2 = ··· = φr = 0, the endomorphism which sends all points in E to the point at inﬁnity. We will assume

that P1,P2,...,Pr are linearly independent over End(E). We should note that each Pn n ∈ Z can be viewed as an element of End(E) by sending P to nP = i=1 P . Notice

that this immediately implies that none of P1,P2,...,Pr are points of ﬁnite order as n can be viewed as an endomorphism by above.

We can now ask the same questions we asked in Chapter 3. Let p ∈ N be a prime of good reduction with respect to E, and recall that there are only ﬁnitely many primes of bad reduction. These bad primes correspond to prime divisors of the discriminant of E (see [63, Theorem VII.5.1]). We note the discriminant of E is the discriminant of a minimal Weierstraß equation deﬁning E, which exists by [63,

Corollary VIII.8.3] since E is deﬁned over Q. Denote by E(Fp) the reduction of E

modulo p and let Γp be the reduction of Γ modulo p. Deﬁne index of Γ modulo p as the following integer:

iΓ(p) = [E(Fp):Γp]. CHAPTER 4. ELLIPTIC CURVE ANALOGUES 63

4.1 Statement of Theorems

We now consider the problem of determining whether the following asymptotic relations hold: X ω(iΓ(p)) ∼ cE(Γ)li(x) p≤x

X 0 Ω(iΓ(p)) ∼ cE(Γ)li(x) p≤x and

X 00 log(iΓ(p)) ∼ cE(Γ)li(x), p≤x 0 00 where cE(Γ), cE(Γ), and cE(Γ) are constants that depend on at most E and Γ. √ Since |E(Fp) − p − 1| ≤ 2 p (see Hasse’s Theorem [73, Theorem 4.4.2] or [63,

Theorem V.1.1]) and p ≥ 2, we have |E(Fp)| ≤ 2x. Therefore, we have the following identities: X X ω(iΓ(p)) = πq(x) p≤x q≤2x X X Ω(iΓ(p)) = πqα (x) p≤x qα≤2x and X X log(iΓ(p)) = Λ(d)πd(x), p≤x d≤2x where

πd(x) := #{p ≤ x : d|iΓ(p)}.

In order to use these equalities we need to show that d|iΓ(p) is equivalent to a Cheb- otarev condition. We note that the only d we need to worry about are prime powers CHAPTER 4. ELLIPTIC CURVE ANALOGUES 64

as Λ(d) is supported on the prime powers, and clearly the other functions are supported on the primes and prime powers, respectively. The primes themselves do not cause any problems. However, for r > 1, there exist some diﬃculties with the prime powers, which we will discuss in the next section. Throughout this chapter, we will use Artin’s Holomorphy Conjecture, which states the following:

Conjecture (Artin’s Holomorphy Conjecture). Let G be a Galois group of a Galois ﬁeld extension L/K of number ﬁelds. Let χ be a non-trivial irreducible character of G. Then, the Artin L-function associated to χ and L/K, denoted L(s, χ, L/K), extends to an entire function of s.

Artin’s Holomorphy Conjecture will be denoted by AHC. See Murty [49] for more details.

Recall that we have an homomorphism from Z to End(E). If this homomorphism is surjective, then we say that E does not have complex multiplication. If this homomorphism is not surjective, then we say that E has complex multiplication. Throughout this chapter, we will write E is CM or non-CM if E has complex multiplication or does not have complex multiplication, respectively. For more in- formation see [63, 64, 65, 73]. We say AHC holds for E if it holds for all the ﬁelds

−1 −1 −1 L = Km = Q(E[m], m P1, m P2, . . . , m Pr) over K = Q as m ranges over N.

Deﬁne c1 and c3 as follows:   r + 3 if E is non-CM   c1 = r+3 , 2 if E is non-CM and AHC holds for E    r+1  2 if E is CM CHAPTER 4. ELLIPTIC CURVE ANALOGUES 65

and   3 if E is non-CM   c3 = 3 if E is non-CM and AHC holds for E .  2   1 if E is CM Let α := 1 and β := r+4 . Deﬁne γ and g as follows: 2(c1+2) 2((c3+2)r+2)  if E is non-CM and r = 15, if E is non-CM, AHC  1 (c3 + 1)β + 2  holds for E, and r ≤ 12, or E is CM and r = 7  γ = ,   if E is non-CM and r ≥ 16, if E is non-CM, AHC (c + 1)α + 1  1 2 holds for E, and r ≥ 13, or E is CM and r ≥ 8 and   if E is non-CM and r = 15, if E is non-CM, AHC 0  holds for E, and r ≤ 12, or E is CM and r = 7  g = .   if E is non-CM and r ≥ 16, if E is non-CM, AHC 1  holds for E, and r ≥ 13, or E is CM and r ≥ 8 In §4.3, we will prove the following theorem:

Theorem 4.1.1. Suppose GRH holds for the Dedekind zeta functions of ﬁelds of type

−1 −1 −1 Km = Q(E[m], m P1, m P2, . . . , m Pr). Also suppose E has rank greater than

4c3 + 2. Then, we have

X γ g ω(iΓ(p)) = cE(Γ)li(x) + O(x (log x) ) p≤x for c3, γ and g deﬁned above and where cE(Γ) is a constant dependent on E and Γ. CHAPTER 4. ELLIPTIC CURVE ANALOGUES 66

Deﬁne c5 as follows:   4 if E is non-CM   c5 = 2 if E is non-CM and AHC holds for E .    1 if E is CM In §4.4, we will prove the following theorems:

Theorem 4.1.2. Let E be an elliptic curve deﬁned over Q. Suppose GRH holds

(i) −1 for the Dedekind zeta functions for ﬁelds of type Lm = Q(E[m], m Pi) for some

i ∈ {1, 2, . . . , r}. Suppose further that r > 4c5 + 2 where c5 is deﬁned above. Then, we have X Ω(iΓ(p)) li(x), p≤x where the implied constant is dependent only on E and Γ.

and

Theorem 4.1.3. Let E be an elliptic curve deﬁned over Q. Suppose GRH holds

(i) −1 for the Dedekind zeta functions for ﬁelds of type Lm = Q(E[m], m Pi) for some

i ∈ {1, 2, . . . , r}. Suppose further that r > 4c5 + 2 where c5 is deﬁned above. Then, we have X log(iΓ(p)) li(x), p≤x where the implied constant is dependent only on E and Γ.

The only impediments in obtaining asymptotic formulæ for Theorems 4.1.2 and 4.1.3 are the GRH and some kind of Chebotarev condition related to a prime power

α dividing [E(Fp):Γp]. That is, q |[E(Fp):Γp] if and only if the Frobenius automorphism associated to p is an element of some conjugacy class (or classes) of the Galois CHAPTER 4. ELLIPTIC CURVE ANALOGUES 67

group of Q(E[m], m−1Γ) over Q. If a Chebotarev condition does exist, the asymptotic formulæ holds assuming GRH. This will be seen in Chapter 7.

4.2 A Divisibility Criterion

We follow Gupta and Murty [27] and Lang and Trotter [44]. Recall that q denotes

a prime number. Let us ﬁrst consider q|iΓ(p) := [E(Fp):Γp]. To do this, we must consider the ﬁeld

−1 −1 −1 Kq := Q(E[q], q P1, q P2, . . . , q Pr)

Pn where E[q] = ker(P 7→ qP ) and nP = i=1 P in the additive Abelian group E(Q) for any n ∈ N. It should be noted, as in Chapter 1, Kq is the smallest ﬁeld containing Q

−1 and the x- and y-coordinates of q Pi for all i ∈ {1, 2, . . . , r}. The elements of E[q] are

−1 called the q-division points of E. If Q0 = q P , then obviously qQ0 = P . However,

we need to ﬁnd all such solutions. We note that if R ∈ E[q], then Q := Q0 + R is

also satisfies qQ = qQ0 + qR = qQ0 = P . Thus, every q-division point gives us a new point Q which satisfies qQ = P , or equivalently, Q = q−1P . To determine how many such solutions exist, we need to determine how many R above there are. Looking at the polynomial equation for qP , we see that q2 − 1 non- trivial such R as well as the trivial solution, the point at infinity (see [63, Exercise 3.7] or [73, Theorem 3.3.2]). This gives us q2 such solutions. We note that this polynomial

equation also shows Kq is Galois over Q.

Deﬁne the action of γ ∈ GL2(Fq) on γ(P1,P2,...,Pr) by (γP1, γP2, . . . , γPr) where

γP for P ∈ E[q] is deﬁned as follows. We will see later on that p|iΓ(p) contributes a small number of elements to our summations. Thus, we assume q 6= p. We know CHAPTER 4. ELLIPTIC CURVE ANALOGUES 68

∼ that E[q] = (Z/qZ) × (Z/qZ). Therefore, there exist Q1,Q2 such that every element of E[q] can be written as some aQ1 + bQ2 for some a, b ∈ Fq := Z/qZ. Thus, we may a a represent P by the column vector ( b ). Hence, γP = γ ( b ). Using theorems of Bashmakov (see [4] or [42, Chapter 5]), it can be shown that

r Gal(Kq/Q) is a subgroup of GL2(Fq) n E[q] . To see this, let’s follow Lang and Trotter [44]. We see, in the case r = 1, that the extension of E[q], the translation

−1 group, by GL2(Fq) operates on q P with P ∈ E(Q) being a non-torsion point. That

−1 is, ﬁx Q0 ∈ q P . Then, σ in the aﬃne group can be represented by (γ, τ) where

γ ∈ GL2(Fq) and τ ∈ E[q]. They satisfy

σQ = (γ, τ)Q = Q0 + γ(Q − Q0) + τ.

From this, we see that Gal(Q(E[q], q−1P )/Q(E[q])) is a subgroup of E[q]. That is, Gal(Q(E[q], q−1P )/Q(E[q])) is a subgroup consisting of translations. In fact, by [4] or [42, Theorem 5.2 of Chapter 5], Gal(Q(E[q], q−1P )/Q(E[q])) = E[q] for all but ﬁnitely many primes q. For the case r > 1, the above reasoning works when we

−1 −1 replace Q0 ∈ q P with λ : hP1,P2,...,Pri → q hP1,P2,...,Pri, a ﬁxed section, satisfying qλP = P for all P ∈ Γ = hP1,P2,...,Pri. The above equation now becomes

σQ = (γ, τ)Q = γ(Q − λQ) + λqQ + τqQ

−1 r for Q ∈ q Γ. From this, Gal(Kq/Q) being a subgroup GL2(Fq)nE[q] is more appar- ent. In fact, Ribet [56] has shown that for suﬃciently large q, we have Gal(Kq/Q(E[q])) is isomorphic to E[q]r via the map 1 1 1 1 1 1 P , P ,..., P 7→ P + a , P + a ,..., P + a q 1 q 2 q r q 1 1 q 2 2 q r r CHAPTER 4. ELLIPTIC CURVE ANALOGUES 69

r where (a1, a2, . . . , ar) ∈ E[q] and assuming E is CM. If E is non-CM, this is still true by the theorems of Bashmakov (see [4] or [42, Theorem 5.2, Chapter 5]). Also, the structure of Gal(Q(E[q])/Q) is well understood from the standpoint of its size and q is suﬃciently large (see [1]).

Let σ ∈ Gal(Kq/Q). Then, we can view σ as a pair (γ, τ) with γ ∈ GL2(Fq) and τ ∈ E[q]r. With this in mind, each σ determines a homomorphism τ :Γ → E[q], and

τ(Γ) is the subgroup of E[q] generated by a1, a2, . . . , ar above. Then, Lang and Trotter [44] proved the following result:

Lemma 4.2.1. Let Cq be the set of elements σ = (γ, τ) of Gal(Kq/Q) such that

(i) ker(γ − 1) = E[q] and τ(Γ) ⊂ E[q] is trivial or cyclic,

(ii) ker(γ − 1) is a non-trivial cyclic group and τ(Γ) ⊂ Im(γ − 1).

For p - q∆, where ∆ is the discriminant of E, q|iΓ(p) if and only if σp ∈ Cq, where

σp is the Frobenius element of p in Gal(Kq/Q).

The discriminant of E is the discriminant of a minimal Weierstraß equation deﬁn- ing E, which exists by [63, Corollary VIII.8.3].

By [44] or [27, Page 36], Cq is a conjugacy closed set of Gal(Kq/Q). That is, Cq is

closed under conjugation by elements of Gal(Kq/Q). As such we can use the effective Chebotarev Density Theorem, subject to the GRH, due to Serre [62, Théorème4] to obtain     |Cq| √ Y #{p ≤ x : q|iΓ(p)} = li(x) + O |Cq| x log |Gq| p x |Gq| p∈P(Kq/Q) where Gq = Gal(Kq/Q) and P(Kq/Q) denotes the set of (finite) primes which ramify in Kq over Q. Also, if, in addition, AHC holds for E, then we have by Murty, Murty, CHAPTER 4. ELLIPTIC CURVE ANALOGUES 70

and Saradha [52] that     |Cq| 1 √ Y #{p ≤ x : q|iΓ(p)} = li(x) + O |Cq| 2 x log |Gq| p x . |Gq| p∈P(Kq/Q) α Let us now consider the condition q |iΓ(p). Let Kqα be the ﬁeld 1 1 1 E[qα], P , P ,..., P . Q qα 1 qα 2 qα r α Then, as above, we have Kqα is Galois over Q. However, equating q |iΓ(p) with a Chebotarev condition is currently more diﬃcult than anticipated. This is why we are led to upper and lower bounds in Theorems 4.1.2 and 4.1.3. Instead, we have the following results of Akbary, Ghioca, and V. K. Murty [2]:

Theorem 4.2.1 (Akbary, Ghioca and Murty). Let E be a non-CM elliptic curve over

Q with discriminant ∆, let P be a point of inﬁnite order in E(Q), and let m > 1 be

−1 an integer. Suppose GRH holds for Lm = Q(E[m], m P ). Then,

#{p ≤ x : p - m∆ and m|[E(Fp): hP i]} li(x) √ ≤ C(E) + O(m4(log log m) x log(mx)), ϕ(m)2 where C(E) is a constant depending only on the elliptic curve E, and the implied constant is dependent only on E. If, in addition, we assume AHC holds for E, then we have

#{p ≤ x : p - m∆ and m|[E(Fp): hP i]} li(x) √ ≤ C(E) + O(m2(log log m)1/2 x log(mx)), ϕ(m)2 where C(E) is a constant depending only on the elliptic curve E, and the implied constant is dependent only on E.

Let Φ be the number ﬁeld analogue of the Euler phi function. For more informa- tion, see Cojocaru and Murty [11, Page 611], Akbary, Ghioca, and Murty [2, Page CHAPTER 4. ELLIPTIC CURVE ANALOGUES 71

383], or Lang [41, Page 126].

Theorem 4.2.2 (Akbary,Ghioca and Murty). Let m ≥ 3. Let E be a CM elliptic

curve over Q with discriminant ∆, and let P be a point of inﬁnite order in E(Q).

−1 Moreover, suppose GRH holds for Lm = Q(E[m], m P ). Then,

#{p ≤ x : p - m∆ and m|[E(Fp): hP i]} aω(m)d(m)2 ≤ C(E) li(x) + O(maω(m)/2d(m)x1/2 log(mx)) Φ(m) where C(E) is a constant depending only on the elliptic curve E, and the implied constant depends only on E.

α Our quantity of interest is #{p ≤ x : q |iΓ(p)}. Clearly, hPii ⊂ hP1,P2,...,Pri =

Γ, and as such, hPii ⊂ hP1, P2,..., Pri = Γp. So, we have

α α #{p ≤ x : q |iΓ(p)} = #{p ≤ x : q |[E(Fp): hP1, P2,..., Pri]}

α ≤ #{p ≤ x : q |[E(Fp): hPii]} for any i ∈ {1, 2, . . . , r}. So this will allow us to develop upper bounds seen in Theorems 4.1.2 and 4.1.3. Finally, the eﬀective Chebotarev Density Theorem allows us to manage small and medium sized primes. So we still need some method for handling the large primes. The following result of Gupta and Murty [27] allows us to do this:

Lemma 4.2.2 (Gupta, Murty). We have

1+ 2 #{p : |Γp| < y} y r .

4.3 An Asymptotic Result

We will ignore the possibility of p|iΓ(p) until the end of this section. CHAPTER 4. ELLIPTIC CURVE ANALOGUES 72

4.3.1 Small Primes

We have X X X X X ω(iΓ(p)) = πq(x) = πq(x) + πq(x) + πq(x), p≤x q≤2x q≤y y

In order to continue, we need estimates on |Cq| and |Gq|. Ribet [56] has shown

∼ r that for E of type CM and for suﬃciently large q, Gal(Kq/Q(E[q])) = E[q] and the other q are 1 independent of q. For non-CM, we can use the theorems of

r Bashmakov [4]. So if q 6= p is suﬃciently large, then |Gal(Kq/Q(E[q]))| = |E[q] | = r 2r ∼ |(Z/qZ × Z/qZ) | = q because E[q] = (Z/qZ) × (Z/qZ) by [63, Corollary III.6.4]. Now, for suﬃciently large q, (see [60, Chapter 4] and [61], or [11, Propositions 3.6 and 3.8]) we have  1 1 q4 1 − 1 − if E is non-CM  q q2 [Q(E[q]) : Q] .  q2 if E is CM CHAPTER 4. ELLIPTIC CURVE ANALOGUES 73

Putting all this together, we obtain

|Gq| = [Kq : Q] = [Kq : Q(E[q])][Q(E[q]) : Q]   2(r+2) q if E is non-CM .  2(r+1) q if E is CM Following Gupta and Murty [27], we have   r+3 r+2 q + O(q ) if E is non-CM |Cq| = .  r+1 r q + O(q ) if E is CM Hence, ! X |Cq| X |Cq| X 1 = + O |G | |G | qr+1 q≤y q q≥2 q q>y 1 = c (Γ) + O E yr

where cE(Γ) is a constant. So, we have     X li(x) √ X Y π (x) = c (Γ)li(x) + O + O x qc1 log qc2 p x q E yr     q≤y q≤y P(Kq/Q) where   r + 3 if E is non-CM   c1 = r+3 2 if E is non-CM and AHC holds for E    r+1  2 if E is CM and   2(r + 2) if E is non-CM c2 = .  2(r + 1) if E is CM CHAPTER 4. ELLIPTIC CURVE ANALOGUES 74

By an argument similar to that of [63, Theorem VII.7.1], the only primes that ramify

in Kq are q and the primes where E has bad reduction. Thus,

Y p q∆.

p∈P(Kq/Q) Hence,     X li(x) √ X Y π (x) = c (Γ)li(x) + O + O x qc1 log qc2 p x q E yr     q≤y q≤y P(Kq/Q) ! li(x) √ X = c (Γ)li(x) + O + O x qc1 log(qx) E yr q≤y √ li(x) yc1+1 x log x = c (Γ)li(x) + O + O E yr log y since q ≤ 2x. In order for these error terms to be o(x/ log x) we need y → ∞ as x → ∞ and √ yc1+1 x log x x = o , log y log x or equivalently, √ yc1+1 x = o . log y (log x)2 We note that this allows us to choose y xα for some α > 0. This will be important in the sequel. Also note that we can now cancel log y and log x.

4.3.2 Intermediate and Large Primes

Let

MΓ(ξ1, ξ2) := #{p ≤ x : q|iΓ(p) for some q ∈ (ξ1, ξ2]}.

We claim X πq(x) MΓ(y, z). y

To see this, note that if p contributes to the above left-hand summation, then there

nα n are q1, q2, . . . , qn ∈ (y, z] prime such that q1q2 ··· qn|iΓ(p). Hence, x y ≤

q1q2 ··· qn ≤ iΓ(p) x since iΓ(p) ≤ |E(Fp)| ≤ 2x. However, since α > 0, nα ≥ 1 for suﬃciently large n. Thus, p can only contribute a bounded (dependent on E) number of times to the above summation. Thus, the claim holds. Therefore, following Gupta and Murty [27], we have X X li(x) √ π (x) M (y, z) ≤ c(E) + O qc3 x log(qx) q Γ q2 yy q≤z √ li(x) zc3+1 x log x c(E) + , y log z where   3 if E is non-CM c3 = .  1 if E is CM Since y xα, we do not need to worry about the ﬁrst term. That is, lix lix c(E) x1−α. y xα For the second error term we need √ zc3+1 x log x x = o , log z log x or equivalently, √ x zc3+1 = o , log x since xα y ≤ z ≤ 2x implies log x log z. By the same argument at the beginning of this section, we see that X πq(x) #{p ≤ x : q|iΓ(p) for some q ∈ (z, 2x]}. z

However, n xo #{p ≤ x : q|i (p) for some q ∈ (z, 2x]} ≤ # p : |Γ | ≤ , Γ p z and by Lemma 4.2.2, we have r+2 2x x r # p : |Γ | ≤ . p z z We need this last term to be o(x/ log x). That is,

2 1+ 2 x r log x = o(z r ).

4.3.3 Optimal Error Terms

We would like to optimize the error terms. Let us ﬁrst record them all: r+2 X li(x) √ li(x) √ x r ω(i (p)) = c (Γ)li(x) + O + yc1+1 x log x + + zc3+1 x + Γ E yr y z p≤x where   r + 3 if E is non-CM   c1 = r+3 2 if E is non-CM, and AHC holds for E    r+1  2 if E is CM and   3 if E is non-CM   c3 = 3 if E is non-CM, and AHC holds for E .  2   1 if E is CM

Let y = xα(log x)A for some α > 0 and A ∈ R. Similarly, let z = xβ(log x)B for some β > 0 and B ∈ R. CHAPTER 4. ELLIPTIC CURVE ANALOGUES 77

For y, we have error terms 1−α c +1√ x (c +1)α+ 1 A(c +1)+1 x y 1 x log x + = x 1 2 (log x) 1 + . y log x (log x)A+1 1 1 1 We want 1 − α = (c1 + 1)α + , and so, (c1 + 2)α = . Hence, α = . No matter 2 2 2(c1+2) what value of A we choose, the above error term is a power saving error term. The

1− 1 optimal choice of A is 0. Thus, these error terms yield x 2(c1+2) log x. For z, we have error terms r+2 r+2 1−β r c +1√ x r (c +1)β+ 1 B(c +1) x z 3 x + = x 3 2 (log x) 3 + . z (log x)B 1 r+2 (c3+2)r+2 r+2 1 r+4 As before, we want (c3 + 1)β + 2 = (1 − β) r . So, r β = r − 2 = 2r . r+4 1 Thus, β = . Also, the optimal choice of B is 0. We want (c3 + 1)β + < 1, 2((c3+2)r+2) 2 or equivalently, β < 1 . Hence, we need 2(c3+1) r + 4 1 < . 2((c3 + 2)r + 2) 2(c3 + 1) Thus,

(c3 + 1)(r + 4) = c3r + r + 4c3 + 4 < c3r + 2r + 2.

Therefore, 4c3 + 2 < r. Thus, we need   15 if E is non-CM   r ≥ 9 if E is non-CM, and AHC holds for E .    7 if E is CM Finally, we would like to calculate which error term is larger, which may be dependent

1 1 on r. Comparing (c3 + 1)β + 2 and (c1 + 1)α + 2 it can be determined that the first term is larger for r = 15 and the second term is larger for all r ≥ 16 if E is non-CM, the first term is larger if r ≤ 12 and the second term is larger for all r ≥ 13 if E is non-CM and AHC holds for E, and the first term is larger for r = 7 and the second CHAPTER 4. ELLIPTIC CURVE ANALOGUES 78

term is larger for all r ≥ 8 if E is CM. These conditions correspond to the conditions in the deﬁnition of γ and g.

To ﬁnish the proof we must deal with #{p ≤ x : p|iΓ(p)}. Now, if p|iΓ(p), then

p + 1 − ap ≡ 0 (mod p) and so ap = 1 for all such p > 5, where ap := p + 1 − |E(Fp)|. Serre [62, Th´eor`eme20] has shown, assuming GRH and E is non-CM, that we have 5 x 6 #{p ≤ x : ap = 1} 1 . (log x) 3 x1/2 Also, for CM elliptic curves, we have the above number is log x (see Cox [13, Theorem 14.16] and Cojocaru [10, Theorem 9]). It should be noted that if we are interested in obtaining o(li(x)), then we do not

need GRH in proving #{p ≤ x : ap = 1} = o(li(x)). However, if we do not assume GRH here, we obtain much worse error terms. This yields the ﬁnal version of the theorem:

Theorem 4.3.1. Suppose GRH holds for the Dedekind zeta functions of ﬁelds of type

−1 −1 −1 Km = Q(E[m], m P1, m P2, . . . , m Pr). Also suppose r > 4c3 + 2 where c3 is deﬁned above. Then, we have

(a) If E is non-CM and r = 13, or E is non-CM, AHC holds for E and r ≤ 12, or E is CM and r = 7, then

1 X (c3+1)β+ ω(iΓ(p)) = cE(Γ)li(x) + O(x 2 ) p≤x

where β and c3 are deﬁned above.

(b) If E is non-CM and r ≥ 16, if E is non-CM, AHC holds for E and r ≥ 13, or if E is CM and r ≥ 8, then we have

1 X (c1+1)α+ ω(iΓ(p)) = cE(Γ)li(x) + O(x 2 log x), p≤x CHAPTER 4. ELLIPTIC CURVE ANALOGUES 79

where α, and c1 are deﬁned above.

4.4 Proofs of Theorems 4.1.2 and 4.1.3

Our goal in this section is to prove Theorems 4.1.2 and 4.1.3. We note that X li(x) ω(ia(p)) p≤x X X ≤ Ω(ia(p)) = πqα (x) p≤x qα≤2x X (log q)πqα (x) qα≤2x X = log(iΓ(p)). p≤x P So, if we can show p≤x log(iΓ(p)) li(x), then these theorems are true. We proceed as in the previous section and use Theorems 4.2.1 and 4.2.2. Also, as in the last

α section, we will not concern ourselves with p |iΓ(p) until the end of the proof. We have X X X X log(iΓ(p)) = (log q)πqα (x) + (log q)πqα (x) + (log q)πqα (x) p≤x q≤y y

X X α (log q)πqα (x) = (log q)#{p ≤ x : q |iΓ(p) and p - q∆} qα≤y qα≤y

X α ≤ (log q)#{p ≤ x : q |[E(Fp): hP1i] and p - q∆} qα≤y X f(qα)2 log q √ li(x) + O qc5αf(qα)(log q)(log log qα)c6 x log(qαx) E q2α qα≤y CHAPTER 4. ELLIPTIC CURVE ANALOGUES 80

by Theorems 4.2.1 and 4.2.2, where   4 if E is non-CM   c5 = 2 if E is non-CM, and AHC holds for E ,    1 if E is CM   1 if E is non-CM   c6 = 1 if E is non-CM, and AHC holds for E  2   0 if E is CM, and  1 if E is non-CM   α  f(q ) = 1 if E is non-CM, and AHC holds for E .    α d(q ) = α + 1 if E is CM Now, we have X f(qα)2 log q X 1 X 1 1 q2α q2α−ε m2−ε qα≤y qα≥1 m≥1 since f(qα) qαε for all ε > 0. We will attack the other terms depending on whether E is non-CM or CM. If E is non-CM, then we have X qc5αf(qα)(log q)(log log qα)c6 x1/2(log qαx) yc5+1x1/2(log x)2(log log x)c6 . qα≤y If E is CM, then we have X qc5αf(qα)(log q)(log log qα)c6 x1/2(log qαx) yc5+1x1/2(log x)3(log log x)c6 qα≤y CHAPTER 4. ELLIPTIC CURVE ANALOGUES 81

by the Titchmarsh divisor problem (see [12, Theorem 9.3.1] or [6, Corollary 1]). In order for these terms to be x/ log x we need x1/2 yc5+1 . (log x)4(log log x)c6 Thus, we are going to choose

x1/2(c5+1) y := (log x)C where C > 4 is arbitrary. For large primes, we use Lemma 4.2.2. That is,

α α #{p ≤ x : q |iΓ(p) for some q > z} ≤ #{p ≤ x : iΓ(p) ≥ z} 2x ≤ # p : |Γ | < p z r+2 x r . z We need this term to be x/(log x)2. Thus,

2 2 r+2 x r (log x) z r , and hence

2 2r z x r+2 (log x) r+2 .

Choose

2 D z := x r+2 (log x)

2r where D > r+2 is arbitrary.

Note that if r ≥ 4c5 + 2, then we have no intermediate primes to check as z ≤ y and the middle summation will be zero. So suppose r < 4c5 + 2. To manage the intermediate primes we will proceed as in the previous section. CHAPTER 4. ELLIPTIC CURVE ANALOGUES 82

We have

X X α (log q)πqα (x) ≤ log x #{p ≤ x : p - q∆ and q |iΓ(p)} yy qα≤z x + zc5+1x1/2(log x)2(log log x)c6 . y1−ε Now our choice of y gives that the ﬁrst term is o(xδ) for some δ < 1. The other term becomes

2c +2 c +1 1/2 2 c 5 + 1 D+2 c z 5 x (log x) (log log x) 6 = x r+2 2 (log x) (log log x) 6

4c5+r+6 = x 2(r+2) (log x)D+2(log log x)c6 .

Now, it can be shown that the exponent is less than 1 if and only if r > 4c5 + 2.

α α α To ﬁnish, we need to consider #{p ≤ x : p |iΓ(p)}. If p |iΓ(p), then p|iΓ(p) and

so as before, ap = 1. Thus, we have

α α X #{p ≤ x : p |iΓ(p)} = 1 pα≤x α p |iΓ(p) X log x ≤ log p p≤x p|iΓ(p)

5 2 x 6 (log x) 3 .

Also, for CM elliptic curves, we again have the above number is x1/2. Thus, our theorem is complete. Again, it should be noted that the GRH is not needed to prove o(li(x)). However, it leaves with us signiﬁcantly worse error terms and the GRH is needed for the beginning CHAPTER 4. ELLIPTIC CURVE ANALOGUES 83

of the proof. Also, it should be noted that assuming a Chebotarev condition is discovered, this proof can be used to obtain signiﬁcantly better error terms. Chapter 5

Summations related to the Artin’s Conjecture

5.1 Generalizing the Titchmarsh Divisor Problem

Recall that the Titchmarsh Divisor Problem [12, §9.3] is interested in evaluating the following summation: X d(p − a) p≤x where a is a ﬁxed non-zero integer. In fact, [6, Corollary 1] gives us X x d(p − a) = cx + c li(x) + O 1 (log x)A p≤x where c and c1 are eﬀectively computable constants dependent on a and the implied constant depends only on a and A. We wish to consider X p − 1 d k p≤x p≡1 mod k 84 CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 85

where k ∈ N is ﬁxed. Let   √ 1 if n ∈ Z δ(n) := .  0 otherwise So, we have X d(n) = 2 1 + δ(n).

d|√n d< n Then, we have X p − 1 X X X p − 1 d = 2 1 + δ k √ k p≤x p≤x d≤ p−1 p≤x p≡1 mod k p≡1 mod k k p≡1 mod k p−1 d| k ! X X = 2 π(x; kd, 1) + O δ(n) . √ x−1 n≤x d≤ k Now,

X 2 δ(n) = #{n ≤ x : n = m for some m ∈ N} n≤x √ = #{m2 ≤ x} ≤ x.

So, we have X p − 1 X √ d = 2 π(x; kd, 1) + O x . k √ p≤x d≤ x−1 p≡1 mod k k CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 86

However, X X li(x) li(x) π(x; kd, 1) = π(x; kd, 1) − + √ √ ϕ(kd) ϕ(kd) x−1 x−1 d≤ k d≤ k  

X 1 X li(x) = li(x) + O  π(x; kd, 1) −  √ ϕ(kd)  √ ϕ(kd)  x−1 x−1 d≤ k d≤ k  

X 1 X li(x) = li(x) + O  π(x; d, 1) −  √ ϕ(kd) √ ϕ(d) x−1 d≤ kx d≤ k X 1 x(log k)2(log log x)B = li(x) + O . √ ϕ(kd) (log x)2 x−1 d≤ k We have used [7, Main Theorem]. So, we just need to deal with X 1 . ϕ(kd) d≤x We will evaluate this summation by using [12, Exercises 6.3 and 6.4] as a guide. We have X kd X X µ2(w) X µ2(w) X X µ2(w) X = = 1 = 1 ϕ(kd) ϕ(w) ϕ(w) ϕ(w) d≤x d≤x w|kd w≤kx d≤x w≤kx d≤x w w|kd gcd(w,k) |d ! X µ2(w) x gcd(w, k) X µ2(w) gcd(w, k) X 1 = = x + O ϕ(w) w wϕ(w) ϕ(w) w≤kx w≤kx w≤kx ! X µ2(w) gcd(w, k) X µ2(w) gcd(w, k) = x + O x + O (log(kx)) . wϕ(w) wϕ(w) w≥1 w>kx However, X µ2(w) gcd(w, k) X µ2(w) A k log(kx) 1 ≤ k = k + O . wϕ(w) wϕ(w) kx (kx)2 x w>kx w>kx CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 87

So ! X kd X µ2(w) gcd(w, k) X µ2(w) gcd(w, k) = x + O x + O (log(kx)) ϕ(kd) wϕ(w) wϕ(w) d≤y w≥1 w>kx X µ2(w) gcd(w, k) = x + O(log(kx)). wϕ(w) w≥1 We notice that X µ2(w) gcd(w, k) X µ2(w) ≤ k , wϕ(w) wϕ(w) w≥1 w≥1 which is a constant dependent on k. We deﬁne X µ2(w) gcd(w, k) c := . k wϕ(w) w≥1 Thus, we have X kd = c x + O(log(kx)). ϕ(kd) k d≤x So, we have X 1 1 X kd = . ϕ(kd) k dϕ(kd) d≤x d≤x However, X kd 1 X kd Z x 1 X kd = + du dϕ(kd) x ϕ(kd) u2 ϕ(kd) d≤x d≤x 1 d≤u Z x Z x log(kx) ck log(ku) = ck + O + du + O 2 du x 1 u 1 u

= ck log x + O(log k) CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 88

since X µ2(w) gcd(w, k) Y 1 Y p2(p − 1) c = = 1 + k wϕ(w) p(p − 1) (p − 1)(p2 − p + 1) w≥1 p p|k Y p − 1 Y 1 1 + ≤ 1 + p(p − 1) + 1 p − 1 p|k p|k X µ2(d) X 1 = log k. ϕ(d) ϕ(d) d|k d≤k So, we have X 1 1 ck log k = (c log x + O(log k)) = log x + O . ϕ(kd) k k k k d≤x Thus, X X 1 x(log k)2(log log x)B π(x; kd, 1) = li(x) + O √ √ ϕ(kd) (log x)2 x−1 x−1 d≤ k d≤ k c c log k x log k x(log k)2(log log x)B = k x − k li(x) + O + O 2k 2k k log x (log x)2 c x(log k)(1 + c ) x(log k)2(log log x)B = k x + O k + O 2k k log x (log x)2 Thus, 2 B X p − 1 ck x(log k)(1 + ck) x(log k) (log log x) d = x + O + O . k k k log x (log x)2 p≤x p≡1 mod k Therefore, we have the following theorem:

Theorem 5.1.1. For any k ∈ N, k > 1 we have X 1 ck log k = log x + O ϕ(kd) k k d≤x and 2 B X p − 1 ck x(log k)(1 + ck) x(log k) (log log x) d = x + O + O k k k log x (log x)2 p≤x p≡1 mod k CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 89

where X µ2(w) gcd(w, k) Y 1 Y 1 c = = 1 + 1 + k wϕ(w) p − 1 p(p − 1) w≥1 p|k p-k and the O constants are absolute.

1 −ε Choose ε > 0. If we assume k < x 10 , then, by Theorem 9 of [6], we have X X 1 x π(x; kd, 1) = li(x) + O . √ √ ϕ(kd) (log x)A x−1 x−1 d≤ k d≤ k Thus, we have the following theorem:

Theorem 5.1.2. Let ε > 0 and A > 0. Let k ∈ N with k > 1. Let x ∈ R>0 such that

1 −ε k < x 10 . Then, X p − 1 ck x(log k)(1 + ck) x d = x + O + O k k k log x (log x)A p≤x p≡1 mod k

where ck is deﬁned as above and the O constants are dependent only on A and ε.

5.1.1 Application to a Generalization of Artin’s Conjecture

Recall, for p - a, we have

d fa(p) = min{d ∈ N : a ≡ 1 mod p}.

Within this section, we also deﬁne fa(p) = ∞ for p|a. This is done for convenience. We want to consider

X 1 X ia(p) = . f (p) p − 1 p≤x a p≤x Let us consider the following summation: 1 X X 1 . y f (p) a≤y p≤x a CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 90

We will also see that this summation is related to the Titchmarsh divisor problem (see [12, §9.3]): X x d(p − a) = cx + c li(x) + O 1 (log x)A p≤x

for any A > 0, where c and c1 are constants, and the implied constant is dependent only on a and A.

y+1 Consider the following: since fkp(p) = ∞ for any k ∈ Z and y = p p − 1 ≤ l y+1 m p p − 1, we have   X X 1 X X 1 X 1 ≤  + ··· +  fa(p)  fa(p) fa(p) p≤x a≤y p≤x 1≤a≤p−1 y+1 y+1 (d p e−1)p+1≤a≤d p ep−1 X y + 1 X 1 = p f (p) p≤x a≤p−1 a

l y+1 m y+1 since fa(p) = fa+kp(p) by deﬁnition. Therefore, since p = p + O(1), we have X X 1 X y + 1 X 1 ≤ + O(1) f (p) p f (p) p≤x a≤y a p≤x a≤p−1 a ! X 1 X 1 X X 1 = (y + 1) + O . p f (p) f (p) p≤x a≤p−1 a p≤x a≤p−1 a Similarly, we have ! X X 1 X 1 X 1 X X 1 ≥ (y + 1) + O . f (p) p f (p) f (p) p≤x a≤y a p≤x a≤p−1 a p≤x a≤p−1 a In particular, ! X X 1 X 1 X 1 X X 1 = (y + 1) + O f (p) p f (p) f (p) p≤x a≤y a p≤x a≤p−1 a p≤x a≤p−1 a ! X 1 X 1 X X 1 = y + O p f (p) f (p) p≤x a≤p−1 a p≤x a≤p−1 a CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 91

Let us consider the error term ﬁrst. We note that the number of elements of

(Z/pZ)∗ that have order k is ϕ(k) provided k|p − 1. So, we have ∗ X 1 X #{a ∈ ( /p ) : fa(p) = k} X ϕ(k) = Z Z = . fa(p) k k a≤p−1 k|p−1 k|p−1 Therefore, X X 1 X X ϕ(k) X = ≤ d(p − 1) x fa(p) k p≤x a≤p−1 p≤x k|p−1 p≤x by the Titchmarsh divisor problem [12, §9.3]. Let us now consider the main term. We ﬁrst note that we have X 1 X 1 X ϕ(k) X 1 = p fa(p) k p p≤x a≤p−1 k≤x p≤x p≡1 mod k since X 1 X ϕ(k) = . fa(p) k a≤p−1 k|p−1 By partial summation, we have X 1 π(x; k, 1) Z x π(u; k, 1) = + du. p x u2 p≤x k p≡1 mod k Therefore, X 1 X 1 X ϕ(k) π(x; k, 1) Z x π(u; k, 1) = + 2 du . p fa(p) k x u p≤x a≤p−1 k≤x k We will evaluate each of these sums separately. The ﬁrst summation is dealt with as follows: X ϕ(k) π(x; k, 1) 1 X 1 X ≤ π(x; k, 1) = d(p − 1) 1 k x x x k≤x k≤x p≤x CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 92

by the Titchmarsh divisor problem [12, §9.3]. In order to evaluate the second summation, notice that X ϕ(k) Z x π(u; k, 1) Z x 1 X ϕ(k) du = π(u; k, 1) du. k u2 u2 k k≤x k 2 k≤u So, in order to evaluate our desired sum, we need to evaluate X ϕ(k) π(x; k, 1). k k≤x We have X ϕ(k) X ϕ(k) X X X ϕ(k) π(x; k, 1) = 1 = k k k k≤x k≤x p≤x p≤x k≤x p≡1 mod k p≡1 mod k X X X µ(d) X X µ(d) X = = 1 d d p≤x k≤x d|k p≤x d|p−1 k| p−1 p≡1 mod k d X µ(k) X p − 1 = d . k k k≤x p≤x p≡1 mod k 1 −ε So, assuming the Theorem 5.1.2 for all k ≤ x instead of k ≤ x 10 , we get X ϕ(k) X µ(k) X p − 1 π(x; k, 1) = d k k k k≤x k≤x p≤x p≡1 mod k X µ(k) ck x(log k)(1 + ck) x = x + O + O k k k log x (log x)A k≤x ! ! X µ(k)ck x X (1 + ck) log k x X 1 = x + O + O k2 log x k2 (log x)A k k≤x k≤x k≤x ! X µ(k)ck x X (1 + ck) log k x = x + O + O . k2 log x k2 (log x)A−1 k≤x k≤x CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 93

Now, we have

X µ(k)ck X µ(k)ck X µ(k)ck = − k2 k2 k2 k≤x k≥1 k>x ∞ ∞ ∞ X µ(k) X µ2(w) gcd(w, k) X µ(k) X µ2(w) gcd(w, k) = − k2 wϕ(w) k2 wϕ(w) k=1 w=1 k>x w=1  ∞  Y 1 X µ(k) Y p2 X µ(k) Y p2 = 1 + − p(p − 1)  k2 p2 − p + 1 k2 p2 − p + 1 p k=1 p|k k>x p|k

 ∞  Y 1 X Y 1 X Y 1 = 1 + µ(k) − µ(k) p(p − 1)  p2 − p + 1 p2 − p + 1 p k=1 p|k k>x p|k

 ∞    Y 1 X Y 1 X Y 1 = 1 + µ(k) + O p(p − 1)  p2 − p + 1  p(p − 1) p k=1 p|k k>x p|k ! Y 1 Y 1 X 1 = 1 + 1 − + O p(p − 1) p2 − p + 1 kϕ(k) p p k>x Y (p(p − 1) + 1)(p2 − p + 1 − 1) 1 = + O p(p − 1)(p2 − p + 1) x p 1 = 1 + O . x So, X ϕ(k) x π(x; k, 1) = x + O . k log x k≤x This is what we expect given §7.1. To see this, we note that given §7.1, it is expected that X x i (p) = c x + c0 lix + O a a a (log x)A p≤x 0 where ca and ca are constants, and ca = 1 on average. Applying partial summation [12, Theorem 1.3.1] to the summation

X 1 X ia(p) = f (p) p − 1 p≤x a p≤x CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 94

would give us

X ia(p) = c log x + O(log log x). p − 1 a p≤x

As we are taking an average of these summations, and ca = 1 on average, we would expect 1 X X 1 = log x + O(log log x). y f (p) a≤y p≤x a 1 −ε So, we need to extend the above work to k ≤ x instead of k < x 10 . This, however, is currently beyond the theory’s potential.

5.2 The number of prime and prime power divisors

of ia(p)

In Chapter 2, we proved the following identities: X x (log(i (p)))α = C π(x) + O a a (log x)2−ε−α p≤x and X x d(i (p)) = C0 π(x) + O a a (log x)2−ε p≤x 0 for any ε > 0, and where Ca and Ca are constant dependent on a. In this section, we will consider the summations X ω(ia(p)) p≤x and X Ω(ia(p)). p≤x Theorem 5.2.1. Let a be a non-zero integer that is not ±1. Suppose GRH holds for CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 95

1/n th the Dedekind zeta functions of Kn := Q(a , ζn) where ζn is a primitive n root of unity as n ranges over primes. Then, X x log log x ω(i (p)) = c li(x) + O a a (log x)2 p≤x

where ca ≥ 0 is a constant depending on a.

Proof. We have X X X X X X ω(ia(p)) = 1 = 1 = πq(x)

p≤x p≤x q|ia(p) q≤x p≤x q≤x q|ia(p) X X X = πq(x) + πq(x) + πq(x) q≤y y

πd(x) := #{p ≤ x : d|ia(p)}.

By the eﬀective Chebotarev Density Theorem and GRH, we have li(x) √ πd(x) = + O( x log(dx)) nd

where nd = [Kd : Q]. Therefore, X X li(x) √ π (x) = + O( x log(qx)) q n q≤y q≤y q √ X 1 y x log x = li(x) + O n log y q≤y q ! √ X 1 X 1 y x log x = li(x) + O li(x) + O . n n log y q≥2 q q>y q

We know that nd dϕ(d). Thus, X 1 c := < ∞ a n q≥2 q CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 96

and X 1 X 1 1 . qϕ(q) q2 y log y q>y q>y Thus, √ X li(x) y x log x π (x) = c li(x) + O + O . q a y log y log y q≤y √ Suppose z > x. Then, we have X X πq(x) = #{p ≤ x : q|ia(p)} q>z q>z [ = # {p ≤ x : q|ia(p)}. q>z

To see this last inequality, note that if the prime p contributes to both πq1 (x) and

πq2 (x), then q1 and q2 divide ia(p). However, if q1 6= q2, then since they are primes, √ √ q1q2|ia(p), but q1 > z ≥ x and q2 > z ≥ x. Thus, x < q1q2 ≤ ia(p) ≤ p − 1 < x, which is a clear contradiction. Therefore, q1 = q2 and the above inequality holds. So, [ {p ≤ x : q|ia(p)} = {p ≤ x : q|ia(p) for some q > z} q>z n xo ⊂ p ≤ x : f (p) ≤ . a z Therefore,  

X n xo  Y m  πq(x) ≤ # p ≤ x : fa(p) ≤ ≤ # p : p (a − 1) z q>z  x  m≤ z X X m ≤ ω(am − 1) x x log m m≤ z m≤ z x2 . z2 log(x/z) Here we have used the result log n ω(n) . log log n CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 97

√ x √ 2 Let y = (log x)4 and z = x(log x) . Then, we have X X √ x x π (x) = c li(x) + π (x) + O x log x + + q a q (log x)3 (log x)5 q≤x y

Then, since q|ia(p) implies q|p − 1, we have X X x X 1 π (x) π(x; q, 1) q log x q y

Similarly, we have the following theorem:

Theorem 5.2.2. Let a be an integer not 0 or ±1. Suppose GRH holds for the

1/n th Dedekind zeta functions of Kn := Q(a , ζn) where ζn is a primitive n root of unity. Then, X x log log x Ω(i (p)) = c0 li(x) + O a a (log x)2 p≤x 0 where ca ≥ 0 is a constant dependent on a.

Proof. We have X X X X X X Ω(ia(p)) = k = 1 = πqk (x) k k p≤x p≤x q kia(p) q≤x p≤x q ≤x k q |ia(p) X X = πqk (x) + πqk (x) qk≤y qk>y CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 98

where y with y ≤ x will be chosen later. As in the previous proof, we have

X X li(x) √ k πqk (x) = + O( x log(q x)) nqk qk≤y qk≤y √ X 1 y x log x = li(x) + O nqk log y qk≤y   √ X 1 X 1 y x log x = li(x) + O li(x)  + O . nqk nqk log y qk≥2 qk>y As before, we have

0 X 1 ca := < ∞ nqk qk≥2 and X 1 X 1 1 . qkϕ(qk) q2k y log y qk>y qk>y Thus, √ X 0 li(x) y x log x π k (x) = c li(x) + O + O . q a y log y log y qk≤y √ x Let y = (log x)4 . Then, X 0 x π k (x) = c li(x) + O . q a (log x)4 qk≤y Now, X X X πqk (x) = πq(x) + πqk (x). qk>y q>y qk>y k≥2 However,

X X x x 3 3 π k (x) √ = x 4 (log x) 2 . q qk y qk>y qk>y k≥2 k≥2 CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 99

Finally, we have X X X πq(x) = πq(x) + πq(x). q>y yz √ Let z = x(log x)2. Then, as in the previous proof, we have X x π (x) q (log x)5 q>z and X x log log x π (x) . q (log x)2 y

5.3 Fomenko’s Result on Average

In this section, we are going to prove the following theorem:

n 1 o Theorem 5.3.1. Assume N > exp c1(log x) 2 for some suﬃciently large constant c1 > 0. Then, ∞ 1 X X X Λ(n) x log(i (p)) = li(x) + O , N a nϕ(n) (log x)D a≤N p≤x n=1 where D is an arbitrary constant greater than 1 and the implied constant is dependent only on D.

We follow Fomenko [20] and Stephens [68, 66]: Deﬁne a character modulo n as

a homomorphism χ :(Z/nZ)∗ → C∗. We can extend such a character χ to all of Z CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 100

(still denoted by χ) by the following rule: χ : Z → C is deﬁned as   χ(a mod n) if gcd(a, n) = 1 χ(a) := .  0 otherwise The principal character is the character associated to the homomorphism χ(a mod n) = 1 for all gcd(a, n) = 1. For a character χ modulo n, and r|p − 1, deﬁne 1 X0 c (χ) = χ(a), r p − 1 a mod p 0 where the indicates that we are summing over a with ia(p) = r. Then, we have the following equalities: 1 X X S := log(i (p)) N a a≤N p≤x 1 X X X X = log w c (χ)χ(a) N w a≤N p≤x w|p−1 χ mod p 1 X X X X = log w c (χ) χ(a) N w p≤x w|p−1 χ mod p a≤N We need the following lemma of Stephens:

Lemma 5.3.1 (Stephens). If χ 6= χ0 the principal character, then gcd(r, k) |c (χ)| ≤ . r rk Also, ϕ ((p − 1)/r) c (χ ) = . r 0 p − 1 CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 101

This gives us 1 X X ϕ((p − 1)/w) [N] S = (log w) [N] − N p − 1 p p≤x w|p−1   1 X X X X + (log w) |c (χ)| χ(a) N w  p≤x w|p−1 χ6=χ0 a≤N X X ϕ((p − 1)/w) li(x) = (log w) + O + O((log x)2 log log x) p − 1 N p≤x w|p−1   1 X X X gcd(ordχ, w) X + O (log w) χ(a) N w ordχ  p≤x w|p−1 χ6=χ0 a≤N Also, the last error term is evaluated in the same way as Fomenko [20]. Since N >

n 1 o exp c1(log x) 2 , we have X X ϕ((p − 1)/w) x S = (log w) + O p − 1 (log x)D p≤x w|p−1 where D is an arbitrary real number greater than 1 and the implied constant is dependent only on D. So we need to evaluate X X ϕ((p − 1)/w) S := (log w) . 1 p − 1 p≤x w|p−1 Fix w ∈ N. Let’s ﬁrst evaluate the following summation: X ϕ((p − 1)/w) T := . (p − 1)/w p≤x w|p−1 CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 102

We have X X µ(d) X µ(d) X µ(d) T = = π(x; wd, 1) = π(x; wd, 1) d d d p≤x d| p−1 d≤ x−1 d≤x w|p−1 w w X µ(d) X µ(d) = π(x; wd, 1) + π(x; wd, 1) d d d≤(log x)B (log x)B (log x)B X µ(d) x = li(x) + O . dϕ(wd) (log x)B d≤(log x)B Now, deﬁne X µ(d) C := . w dϕ(wd) d≥1 This is a constant dependent on w. So,   X µ(d) X 1 = C + O dϕ(wd) w  dϕ(wd) d≤(log x)B d>(log x)B 1 = C + O . w ϕ(w)(log x)B Hence, we have the following lemma:

Lemma 5.3.2. For w ∈ N ﬁxed, we have X ϕ((p − 1)/w) x = C li(x) + O , (p − 1)/w w (log x)B p≤x w|p−1 where X µ(d) C := w dϕ(wd) d≥1 and the implied constant is absolute. CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 103

Finally, note that X X ϕ((p − 1)/w) S := (log w) 1 p − 1 p≤x w|p−1 X log w X ϕ((p − 1)/w) X log w x = = C li(x) + O w (p − 1)/w w w (log x)B w≤x p≤x w≤x w|p−1 ! X X µ(d) log w X X log w x = li(x) + O li(x) + O wdϕ(wd) wdϕ(wd) (log x)B w≥1 d≥1 w≥x d≥1 ! X X µ(d) log w X log w x = li(x) + O li(x) + O wdϕ(wd) wϕ(w) (log x)B w≥1 d≥1 w≥x X X µ(d) log w x = li(x) + O . wdϕ(wd) (log x)B w≥1 d≥1 So, in order to prove the theorem, we need to show X Λ(n) X X µ(d) log w = . nϕ(n) wdϕ(wd) n≥1 d≥1 w≥1 However, X X µ(d) log w X 1 X = µ(d) log w wdϕ(wd) nϕ(n) d≥1 w≥1 n≥1 wd=n X Λ(n) = nϕ(n) n≥1 by M¨obius inversion. A similar analysis will yield the following result:

n 1 o Theorem 5.3.2. Assume N > exp c2(log x) 2 for some suﬃciently large constant c2 > 0. Then, !2 1 X X x2 log(i (p)) − Cli(x) , N a (log x)D a≤N p≤x where D is an arbitrary constant greater than 2, the implied constant is dependent on CHAPTER 5. SUMMATIONS RELATED TO THE ARTIN’S CONJECTURE 104

at most D, and X Λ(n) C = . nϕ(n) n≥1 We immediately have the following corollary:

n 1 o Corollary 5.3.1. Assume N > exp c2(log x) 2 for some suﬃciently large constant c2 > 0. Then, ( ) X N # a ≤ N : log(i (p)) − Cli(x) > εli(x) , a ε2(log x)D−2 p≤x where D > 2 is arbitrary and C is as above.

This corollary is reminiscent of Tur´an’sproof of a Hardy-Ramanujan Theorem about the normal order of ω(n). See [69, 28]. We note that these theorems also apply to the summations

X n (log(ia(p))) . p≤x To see this we just need to change log w and (log w)2 of the above theorems to (log w)n and (log w)2n, respectively. Chapter 6

A Conjecture of Erd˝osand Fomenko’s Theorem

6.1 Erd˝os’s Conjecture

Recall, for p - a, that the order of a modulo p is deﬁned as

n fa(p) := min{n ∈ N : a ≡ 1 mod p} = |ha mod pi|.

Let

δ Aa(x, δ) := #{p ≤ x : fa(p) > p } and

Ea(r) := #{p : fa(p) = r}.

105 CHAPTER 6. A CONJECTURE OF ERDOS˝ AND FOMENKO’S THEOREM106

In 1971, Bundschuh [36] proved the following: For δ < 1/2, r log 2 E (r) ≤ 2 log r x x A (x, δ) = + o 2 log x log x and 1 x x A x, ≥ (1 − log 2) + o . 2 2 log x log x Bundschuh’s techniques can easily be extended to show r log a E (r) ≤ a log r and x x A (x, δ) = + o a log x log x

assuming δ < 1/2. However, his techniques cannot be extended to Aa(x, 1/2). In 1976, Erd˝os[16] improved upon this result. He was able to show:

Theorem 6.1.1 (Erd˝os). As x → ∞, we have 1 x x A x, = + o . 2 2 log x log x

In order to prove this, Erd˝osneeded a better result for E2(r). He showed 1 r log 2 E (r) ≤ + o(1) (6.1) 2 2 log r as r → ∞. Unlike Bundschuh’s methods, Erd˝os’stechniques can be used to prove 1 x x A x, = + o a 2 log x log x and 1 r log 2 E (r) ≤ + o(1) . a 2 log r CHAPTER 6. A CONJECTURE OF ERDOS˝ AND FOMENKO’S THEOREM107

In [16], Erd˝osmade many conjectures related to fa(p). The following conjecture is of interest in this chapter:

Conjecture (Erd˝os’sConjecture). For every ε > 0, there exists a constant C :=

ε C(ε) > 0 and an r0 := r0(ε) ≥ 1 such that E2(r) ≤ Cr for all r ≥ r0.

ε That is, for every ε > 0, E2(r) ε r , where the ε may become for convenience. Throughout this chapter, Erd˝os’sConjecture will be the above with 2 replaced by a ∈ Z. In [19], the author has proved the following theorems:

Theorem 6.1.2 (Felix). Suppose Erd˝os’sconjecture holds for non-zero a ∈ Z with a 6= ±1 or a square. Then, there exist inﬁnitely many primes p for which a is a primitive root modulo p.

x In fact, it was shown that there are at least (log x)2 primes p ≤ x where a is primitive root modulo p. We call the following statement the quasi-Riemann hypothesis: For K an

algebraic number ﬁeld with Dedekind zeta-function ζK (s), there exists some ε ∈

(0, 1/2) such that if <(s) > 1 − ε, then ζK (s) 6= 0.

Theorem 6.1.3 (Felix). Let a ∈ Z diﬀerent from 0, ±1 or a perfect square. Suppose

1/n the quasi-Riemann Hypothesis holds for all Kummer ﬁelds Q(ζn, a ) where ζn is a primitive nth root of unity. Suppose Erd˝os’sconjecture holds for a. Then, x x log log x N (x) = A(a) + O . a log x (log x)2

In §6.2, we will prove the following theorem, which is similar to Theorem 6.1.3: CHAPTER 6. A CONJECTURE OF ERDOS˝ AND FOMENKO’S THEOREM108

Theorem 6.1.4. Let a ∈ Z diﬀerent from 0 or ±1. Suppose the quasi-Riemann

1/n th hypothesis is true for all Kummer ﬁelds Q(ζn, a ) where ζn is a primitive n root of unity. Suppose further that Erd˝os’sconjecture holds for a. Then,

X 1− ε +θ log(ia(p)) = cali(x) + O(x 2 ) p≤x

where ca is an eﬀectively computable constant, ε ∈ (0, 1/2) is ﬁxed such that if <(s) >

1 − ε, then ζK (s) 6= 0, and θ > 0 is arbitrary.

6.2 Proof of Theorem 6.1.4

We have X X X X log(ia(p)) = Λ(d)πd(x) = Λ(d)πd(x) + Λ(d)πd(x) p≤x d≤x d≤xδ xδ

πd(x) := #{p ≤ x : d|ia(p)}.

Now, the quasi-Riemann hypothesis implies

li(x) 1−ε πd(x) = 1/n + O(x log(dx)). [Q(ζn, a ): Q] 1/n We know that [Q(ζn, a ): Q] dϕ(d). We also know that X Λ(d) y d≤y by the Prime Number Theorem or Chebychev’s Theorem. Thus,   X X Λ(d) 1−ε X Λ(d)πd(x) = li(x) 1/n + O x log x Λ(d) [ (ζn, a ): ] d≤xδ d≤xδ Q Q d≤xδ

1−δ 1−ε+δ = cali(x) + O x + O x log x . CHAPTER 6. A CONJECTURE OF ERDOS˝ AND FOMENKO’S THEOREM109

We note that

X 1−δ πq(x) #{p ≤ x : fa(p) < x }. xδ

δ there exist q1, q2, . . . , qn ∈ (x , x] such that qi|ia(p). Hence,

nδ x < q1q2 ··· qn ≤ ia(p) < x.

Thus,

X 1−δ πq(x) #{p ≤ x : fa(p) < x }. xδ

X X log q X log q 1−δ (log q)π α (x) ≤ x x x . q qα q2 xδxδ qα≥xδ α≥2 α≥2 α≥2 θ Now, Erd˝os’sconjecture says #{p : fa(p) = r} r for any θ > 0. Thus, X X X Λ(d)πd(x) = (log q)πq(x) + (log q)πqα (x) xδ

X 1−δ log x πq(x) + O(x ) xδ

1−δ 1−δ log x#{p ≤ x : fa(p) < x } + O(x )

X 1−δ log x Ea(r) + O(x ) r

for any Θ > 1 − δ. Thus,

X 1−δ+θ Λ(d)πd(x) x xδ

for any θ > 0. Therefore,

X 1− ε +θ log(ia(p)) = cali(x) + O(x 2 ) p≤x upon choosing δ = ε/2. Chapter 7

Further Research

7.1 A Second Conjecture of Fomenko

Let a ∈ Z be a squarefree integer diﬀerent from 0 and ±1. In this section, we will consider a second conjecture about ia(p) mentioned by Fomenko [21]: X ia(p) ∼ cax p≤x as x → ∞ where ca > 0 is some constant dependent on a. Deﬁne   2 if 2a|d and a ≡ 1 mod 4 ε(d) := .  1 otherwise

dϕ(d) Hooley [31] and Wagstaﬀ [72, Proposition 4.1] computed [Kd : Q] as ε(d) . With

πd(x) := #{p ≤ x : d|ia(p)} as before, the eﬀective Chebotarev Density Theorem along the GRH give us li(x) √ πd(x) = + O( x log(dx)). [Kd : Q]

111 CHAPTER 7. FURTHER RESEARCH 112

We are now ready to compute conjectural asymptotics for X ia(p). p≤x We have X X X li(x)ϕ(d) √ ia(p) = ϕ(d)πd(x) = + O( x log(dx)) [Kd : ] p≤x d≤x d≤x Q ! X ε(d) √ X = li(x) + O x log x ϕ(d) . d d≤x d≤x At this point we note that if instead of summing d up to x we sum up to z = xα

1 where 0 < α < 4 , then we would get X ia(p) x. p≤x However, for larger values, the error term dominates the main term. So, we are going to assume that this error term is suﬃciently negligible and proceed. That is, X X ε(d) i (p) = li(x) + Negligible error. a d p≤x d≤x So, we just need to compute X ε(d) . d d≤x We have two cases to consider: Case 1: a 6≡ 1 (mod 4).

Then, ε(d) = 1 for all d ∈ N. So, we have X 1 1 = log x + γ + O . d x d≤x Thus, we have X 1 i (p) = li(x) log x + γ + O + Negligible error a x p≤x

= x + γli(x) + Negligible error. CHAPTER 7. FURTHER RESEARCH 113

Case 2: a ≡ 1 (mod 4). Then, we have X ε(d) X 1 X 1 = + 2 d d d d≤x d≤x d≤x 2a1-d 2a1|d X 1 X 1 = + d d d≤x d≤x 2a|d 1 X 1 = log x + γ + O + . x d d≤x 2a|d So, we just need to evaluate this last summation. We have X 1 1 X 1 = d 2|a| d d≤x x d≤ 2|a| 2a|d 1 x 2|a| = log + γ + O 2|a| 2|a| x log x log(2|a|) γ 1 = − + + O . 2|a| 2|a| 2|a| x Thus, we have X 1 γ log 2|a| i (p) = 1 + x + γ + − li(x) + Negligible error. a 2|a| 2|a| 2|a| p≤x We note that in both of these cases we have an additional li(x) term. So if the above formulæ are correct, then this will explain why the average of the following summation has a log log x term as seen in [19]:

X ia(p) X 1 = . p − 1 f (p) p≤x p≤x a We also note that if this conjecture is true, then this will allow us to push α < 1 in the result of Chapter 2 to much larger α. CHAPTER 7. FURTHER RESEARCH 114

P 7.2 Diﬃculties with p≤x log(ia(p))

We recall the original problem of Fomenko [21]: is there a constant ca dependent on a such that X log ia(p) ∼ caπ(x)? p≤x Fomenko was able to show that this is the case assuming GRH and Conjecture A of Hooley. The diﬃculty in proving this result without Conjecture A of Hooley arises

√ 4 √ 2 from the sets {p ≤ x : q|ia(p)} where q is a prime in the range ( x/(log x) , x(log x) ]. A potential plan of attack for this problem is as follows. Recall the following theorem of Fomenko [21]:

Theorem 7.2.1 (Fomenko). Let w ∈ N be ﬁxed and a be a squarefree integer which is not 0 or ±1. Assume GRH holds for Dedekind zeta functions of Kummer ﬁelds

1/n th Kn := Q(a , ζn) where ζn is a primitive n root of unity. Deﬁne

Na(x; w) := #{p ≤ x : ia(p) = w}.

Then, we have x log log x x N (x; w) = A (w)li(x) + O + O , a a ϕ(w)(log x)2 (log x)B where B > 1 is an arbitrary real number, the implied constant is dependent only on a and B, and ∞ X µ(k) Aa(w) = . [Kn : ] k=1 Q Of the above terms, only the last error term causes diﬃculties. To see this, we CHAPTER 7. FURTHER RESEARCH 115

note that the summation in which we are interested is X (log q)#{p ≤ x : q|ia(p)} √ √ x 2 4 a (log x)4 √ x √ ≤ (log x)# p ≤ x : i (p) ∈ , x(log x)2 a (log x)4

√ 2 + (log x){p ≤ x : ia(p) > x(log x) }.

√ 2 The quantity #{p ≤ x : ia(p) > x(log x) } can be bounded by techniques of Hooley [31]. However, the other quantity is too diﬃcult to suﬃciently bound using current techniques. To see this, we note that √ x √ 2 X # p ≤ x : ia(p) ∈ 4 , x(log x) = Na(x; w) (log x) √ √ x

The ﬁrst two terms are bounded by o(li(x)) since Aa(w) 1/wϕ(w) and X 1 log x = A log x + B + O . ϕ(n) x n≤x However, this last term causes clear diﬃculties since it can be bounded by at best x3/2(log x)2−B. This is of a similar order of magnitude for the upper bound of

P 3/2 −1 p≤x ia(p) x (log x) that Fomenko [21] gives. It may be possible to improve this ﬁrst bound by improving Fomenko’s aforementioned theorem about Na(x; w), but current techniques would require a much stronger version of the GRH. CHAPTER 7. FURTHER RESEARCH 116

7.3 Other Generalizations and Potential Improve-

ments

In this section, we will discuss further generalizations and possible improvements of the theorems that are throughout this document.

7.3.1 Elliptic Curves

In Chapter 4, we discussed a generalization of Artin’s conjecture for primitive roots to elliptic curves. However, in some cases we were unable to prove asymptotic

α formulæ because of a missing Chebotarev condition associated to q |iΓ(p). It should be noted that if such a condition is true, then the asymptotic formulæ follow from the proofs given in Chapter 4. To see this, we note that the only prime powers with which we had diﬃculty were the small prime powers (qα ≤ y). If we have

α a Chebotarev condition on q |iΓ(p) along with the generalized Riemann hypothesis, then we have

|Sqα | √ α α li(x) + O(|Sqα | x log(q ∆x) = #{p ≤ x : q |iΓ(p)} |Gqα |

α ≤ #{p ≤ x : q |[E(Fp): hP1i]}

|Cqα | √ α = li(x) + O(|Cqα | x log(q ∆x)), |Lqα | where Sqα is the conjugacy set coming from the Chebotarev condition. Thus, for suﬃciently large x, dependent on qα and E, we have

|S α | |C α | q ≤ q . |Gqα | |Lqα | CHAPTER 7. FURTHER RESEARCH 117

Thus,

X |Sqα | X |Cqα | ≤ < ∞. |G α | |L α | qα≥2 q qα≥2 q The rest of the proof is identical. We should also note that in Chapter 4 we did not choose optimal y because we were only proving upper and lower bounds, and so the error may be signiﬁcantly better than o(li(x)).

Towards a Chebotarev Condition

Working towards a resolution of the the above problem, we have the following com-

α ments. We need only consider when q |iΓ(p) with Γ = hP1,P2,...,Pri ⊂ E(Q) and p is of good reduction. We note that we can easily develop a criterion for qα|[A : B] where B ≤ A are ﬁnite Abelian groups.

Applying this to A = E(Fp) and B = Γp gives us a desirable criterion. For example, q|iΓ(p) if and only if there exists a Q/∈ Γ such that qQ ∈ Γ. That is, there exist m1, m2, . . . , mr ∈ {0, 1, . . . , q − 1} not all zero, such that qQ = m1P1 + m2P2 + ··· + mrPr. This last condition is equivalent to the q-division polynomial having a root modulo p, and polynomials having a root modulo p is equivalent to some Chebotarev condition in a normal field extension of the field where the polynomials live, that is, the field in which their coefficients exist. Note that this is

−1 −1 −1 equivalent to saying Q = q m1P1 + q m2P2 + ··· + q mrPr + µ where µ ∈ E[q].

2 Similarly, we can bootstrap this idea to obtain conditions on when q |iΓ(p) and

3 q |iΓ(p), and so on. It should be noted that a key point in this process is that there are only ﬁnitely many choices for the m1, m2, . . . , mr, n1, n2, . . . , ns, where m1, m2, . . . , mr ∈ CHAPTER 7. FURTHER RESEARCH 118

{0, 1, . . . , q − 1} correspond to P1,P2,...,Pr and n1, n2, . . . , ns ∈ {0, 1, . . . , q − 1} correspond to Q1,Q2,...,Qs with Qi+1 ∈/ hΓ,Q1,Q2,...,Qii and qQ ∈ hQ1,Q2,...,Qii. Again, these are all correspond to Chebotarev conditions. There are a few details which need to worked out before this method gives us a resolution of the above theorem.

7.3.2 Other Sequences and Functions

Throughout, we have been studying summations of the flavour: X log(iΓ(p)) p≤x where Γ ⊂ Q is finitely generated by multiplicatively independent integers or Γ ⊂ E(Q) where E is an elliptic curve defined over Q. However, we can vary these summations in some new ways to see if the techniques used throughout apply to other questions. For example, we saw that changing log n to ω(n) or Ω(n) yielded similar results in each of the above cases. We also saw that changing iΓ(p) to |Γp| in the first case yields a similar result (see Corollary 3.1.1). However, there are many other functions of interest. The following is a small list of other functions which may yield similar results to those seen here: ωk(n), Ωk(n),

α ω(ia(p)) r (log ia(p)) , 2 and dk(n) where α > 1 is a real number, r ≥ 1 and k > 1 are positive integer. The general situation is as follows: write X f(n) = g(d), d|n CHAPTER 7. FURTHER RESEARCH 119

which is possible the M¨obiusinversion formula. Then, X X X f(ia(p)) = g(d)

p≤x p≤x d|ia(p) X X = g(d) 1 d≤x p≤x d|ia(p) X = g(d)πd(x). d≤x Applying the techniques seen throughout this document, will allow us to handle this summation for many functions g. In fact, as f is completely determined by g (and vice versa), the growth of g is important in determining any asymptotic relations about these summations.

We can also change the sequence {iΓ(p)}p∈N,prime to other similar sequences. For example, Li [46], and Li and Pomerance [47] have generalized Artin’s conjecture to composite moduli in the following way: let a be an arbitrary integer. Consider the group (Z/nZ)∗, which is rarely cyclic. However, it does have a maximal order with respect to individual elements known as the Carmichael lambda function

k λ(n) := min{k ∈ N : b ≡ 1 mod n for all b ∈ N with gcd(b, n) = 1}.

k For gcd(a, n) = 1, we still have a multiplicative order fa(n) := min{k ∈ N : a ≡

1 mod n}, and so, we deﬁne a to be a primitive root modulo n if fa(n) = λ(n).

∗ We can similarly deﬁne the index of a modulo n as ia(n) = [(Z/nZ) : ha mod ni]. Then, we can ask whether X f(ia(n)) n≤x has an asymptotic behaviour or not where f(n) = ω(n), Ω(n), or log n. We should note that Li [46] and Li and Pomerance [47] have shown that the CHAPTER 7. FURTHER RESEARCH 120

traditional Artin’s conjecture generalized in this way does not have an asymptotic formula. That is, deﬁne

Na(x) := #{n ≤ x : gcd(a, n) = 1, fa(n) = λ(n)}.

Then, for any a ∈ Z, we have N (x) lim inf a = 0, x→∞ x and there exists an infinite set S ⊂ Z, which has density 0 in Z, such that for a∈ / S we have the following assuming GRH: N (x) ϕ(|a|) lim sup a ≥ A , x→∞ x |a| where A is a positive constant. So we may not obtain asymptotic formulæ for f(n) = ω(n), Ω(n), or log n, but upper and lower bounds are still possible. We should also note that Artin’s conjecture has been extended to function fields (polynomial rings over finite fields) [5, 59], Carlitz modules [33], and Drinfel’d modules of rank one [34]. So it may be possible to extend the work seen here to these settings as well.

7.4 Unconditional Lower Bounds

We should note that throughout we have been assuming some version of GRH, ex- cluding parts of §5.1 and §5.3. However, with many of the theorems developed, we can obtain unconditional lower bounds. For example, in Chapter 4, we can obtain unconditional lower bounds for the summation: X log(iΓ(p)) p≤x CHAPTER 7. FURTHER RESEARCH 121

∗ where Γ = ha1, a2, . . . , ari ⊂ Q and a1, a2, . . . , ar are multiplicatively independent integers. In fact, for any non-zero function f : N → C with X f(n) = g(d) d|n and g(n) ≥ 0 for all n ∈ N it is easy to obtain lower bounds. This is true because we can always truncate the following summation for any y ≥ 1: X X X f(ia(p)) = g(d)πd(x) ≥ g(d)πd(x), p≤x d≤x d≤y

where πd(x) := #{p ≤ x : d|ia(p)}. Applying the unconditional effective Chebotarev Density Theorem and choosing y sufficiently small or finite will give us the desired result: X f(ia(p)) li(x) p≤x

unconditionally. This also works for iΓ(p) where Γ is a ﬁnitely-generated subgroup of

Q∗ or E(Q). In particular, for f(n) = log n, ω(n), Ω(n) or d(n), we have unconditional results. However, if g(n) is negative often, then this technique is much more diﬃcult. For example, consider

Na(x; w) = #{p ≤ x : ia(p) = w} x log log x x = A (w)li(x) + O + O a ϕ(w)(log x)2 (log x)A for any A > 0 and w ∈ N ﬁxed, where X µ(m) A (w) = , a [ (ζ , a1/wm): ] m≥1 Q wm Q th where ζwm is a primitive (wm) root of unity. This constant is always greater than or equal to 0 (as Na(x; w) cannot be negative), but there are a and w for which it is

0. In fact, for Γ, a ﬁnitely-generated subgroup of Q∗, determining which Γ will give CHAPTER 7. FURTHER RESEARCH 122

positive constant is difficult. Let S be a finite set of primes. This difficulty is due to the fact that when trying to calculate X µ(m) A (w) = Γ [ (ζ , Γ1/wm): ] m≥1 Q wm Q

X µ(m1) = · CΓ(w) 1/m1 [Q(ζm1 , Γ ): Q] gcd(m1,S)=1

where CΓ(w) is the other summation resulting from letting m = m1m2 with gcd(m1,S) =

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