Course Notes for Analytic Number Theory

Home , Analytic number theory, Cauchy sequence, Dirichlet convolution, Euler product, Geometric progression, Uniform convergence

Analytic Number Theory Notes on a course taught at the University of East Anglia by G R Everest in Spring 1999

July 20, 2020

Dr C Roettger Mathematics Department Iowa State University 463 Carver Hall, 50011 Ames, IA, USA [email protected] https://faculty.sites.iastate.edu/roettger/

1 Contents

1 Elementary Number Theory and Easy Asymptotics 3 1.1 The Log of the Zeta Function ...... 7 1.2 Euler’s Summation Formula ...... 11 1.3 Multiplicative arithmetical functions ...... 15 1.4 Dirichlet Convolution ...... 20

2 The Riemann Zeta Function 23 2.1 Euler Products ...... 23 2.2 Uniform Convergence ...... 25 2.3 The Zeta Function is Analytic ...... 28 2.4 Analytic continuation of the Zeta Function ...... 30

3 The Functional Equation 35 3.1 The Gamma Function ...... 35 3.2 Fourier Analysis ...... 37 3.3 The Theta Function ...... 41 3.4 The Gamma Function Revisited ...... 46

4 Primes in an Arithmetic Progression 54 4.1 Two Elementary Propositions ...... 54 4.2 A New Method of Proof ...... 55 4.3 Characters of Finite Abelian Groups ...... 61 4.4 Dirichlet characters and L-functions ...... 65 4.5 Analytic continuation of L-functions and Abel’s Summation Formula ...... 69

A Proof of Theorem 3.7 73

B Abel’s Limit Theorem 76

2 Figure 1: Level of Diﬃculty of the Course

Riemann zeta function level Riemann hypothesis

Easter vac 6 33 elementary lectures

Analytic Number Theory

1 Elementary Number Theory and Easy Asymp- totics

Recommended text: Tom APOSTOL, ”Introduction to Analytic Number Theory”, 5th edition, Springer. ISBN 0-387-90163-9.

Course content: • Integers, especially prime integers • connections with complex functions How did the subject arise? E. g. Gauss and Legendre did extensive calcula- tions, calculated tables of primes. By looking at these tables, Gauss reckoned that the real function

π(x) := #{p ≤ x : p is prime} grows quite regularly, namely π(x) log(x) lim = 1, (1) x→∞ x

3 although the individual primes are scattered rather irregularly among the natural numbers. Equation (1) is known as the Great Prime Number The- orem - or just THE Prime Number Theorem. The ﬁrst proof of it used the developing theory of complex functions.

Deﬁnition 1.1 In this course, a prime is a positive integer p which admits no divisors except 1 and itself 1. By deﬁnition, 1 is not a prime.

Theorem 1.2 (Fundamental Theorem of Arithmetic - FTA) Every integer n > 1 can be expressed as a ﬁnite product of primes, unique up to order.

Example: 6 = 2 · 3 = 3 · 2. From the FTA follows the Corollary 1.3 There must be infinitely many primes. Proof 1 (Euclid): If there are only finitely many primes, we can list them p1, . . . , pr. Define N := p1 · ... · pr + 1.

By FTA, N can be factorized, so it must be divisible by some prime pk of our list. Since pk also divides p1 · ... · pr, it must divide 1 - an absurdity. 2 Proof 2 (Euler): If there are only finitely many primes p1, . . . pr, consider the product r −1 Y 1 X := 1 − . pk k=1 Note that the product is well defined, since 1 is not a prime and since, by hypothesis, there are only finitely many primes. Now expand each factor into a geometric series: 1 1 1 1 1 = 1 + + 2 + 3 + ... 1 − p p p p

1In Ring theory, primes are defined in a different way, i. e. an element x of a commutative ring R, not a unit, is prime in R iff for all products in R divisible by x, at least one factor must be divisible by x. For positive integers, this amounts to the same as our definition.

4 Put this into the equation for X: 1 1 1 1 1 1 X = 1 + + + + ... · 1 + + + + ... 2 22 23 3 32 33 1 1 1 1 1 1 · 1 + + 2 + 3 + ... ··· 1 + + 2 + 3 + ... 5 5 5 pr pr pr 1 1 1 1 = 1 + + + + + ... 2 3 4 5 X 1 = , n n the harmonic series! But this series diverges, and again we have reached an absurdity. 2

Why exactly does the harmonic series diverge???

Proof 1 (typical year One proof): 1 1 1 1 1 1 1 1 + + + + + + + + ... 2 3 4 5 6 7 8 1 1 1 1 1 1 1 > 1 + + + + + + + + ... 2 4 4 8 8 8 8 1 2 4 = 1 + + + + ... 2 4 8 1 1 1 = 1 + + + + ... 2 2 2 which cleary diverges. 2

Exercise: Try to prove that P 1/n2 diverges using the same technique. Of course, this will not work, since this series converges, but you will see something ”mildly” interesting.

PN 1 R N 1 Proof 2: Compare n=1 n with the integral 1 x dx = log(N). The result is: N X 1 log(N) ≤ ≤ log(N) + 1. (2) n n=1 The ﬁrst inequality in (2) is enough to show the divergence of the harmonic series. But together with the second, we are given information about the

5 Figure 2: The harmonic series as integral of a step function

speed of the divergence! Proof 2 is a typical example of a proof leading for- ward. 2

6 26.1.99 1.1 The Log of the Zeta Function Deﬁnition 1.4 Given two functions f : R → C and g : R → R+, we write f = O(g) as n → ∞ if there exist constants C and x0 such that |f(x)| ≤ Cg(x) for all x ≥ x0. This is used to isolate the dominant term in a complicated expression. Examples: • x3 + 501x = O(x3)

• Any bounded function is O(1), e. g. sin(x) = O(1)

• Can have complex f: eix = O(1) for real x.

PN 1 Thus we can write 1 n = log(N) + O(1). You may also see f = o(g). This |f| means g → 0 as x tends to inﬁnity. The Prime Number Theorem can be written x x π(x) = + o or log(x) log(x) π(x) log(x) = 1 + o(1). x Theorem 1.5 P 1 The series p p diverges. Proof 1 (see Apostol p. 18): By absurdity. Assume that the series converges, i. e. X 1 < ∞. p p P 1 1 Q So there is some N such that p>N p < 2 . Let Q := p≤N p. The numbers 1 + nQ, n ∈ N, are never divisible by primes less than N (because those divide Q). Now consider

∞ !t X X 1 X 1 < = 1. p 2t t=1 p>N t

7 We claim that ∞ ∞ !t X 1 X X 1 ≤ (3) 1 + nQ p n=1 t=1 p>N because every term of the l.h.s. appears on the right at least once (convince yourself of this claim by assuming e. g. N = 11 and ﬁnd some terms in the r.h.s.!) But the series on the l.h.s. of (3) diverges! This follows from the ’limit comparison test’:

an P If for two real sequences an and bn holds → L 6= 0, then an P bn n converges iﬀ n bn does. 1 1 Apply this test with an = 1+nQ , bn = n and L = 1/Q. This absurdity proves the theorem. 2 Proof 2: We will show that X 1 > log log(N) − 1. (4) p p≤N Remark 1.6 In these notes, we will always deﬁne N = {1, 2, 3,...} (excluding 0). If 0 is to be included, we will write N0. Let N := {n ∈ N : all prime factors of n are less than N} Then X 1 Y 1 1 1 = 1 + + + + ... n p p2 p3 n∈N p≤N ! Y 1 = (5) 1 − 1 p≤N p If n ≤ N, then n ∈ N , therefore X 1 X 1 ≤ . n n n≤N n∈N But log(N) is less than the l.h.s., so

−1 X 1 Y 1 log(N) ≤ = 1 − . (6) n p n∈N p≤N

8 Lemma 1.7 For all v ∈ [0, 1/2] holds 1 2 ≤ ev+v . 1 − v

1 Apply this with v = p (note primes are at least 2, so the lemma does apply):

−1 Y 1 Y 1 1 1 − ≤ exp + . (7) p p p2 p≤N p≤N

Now combine this with equation (6) and take logs:

X 1 1 log log(N) ≤ + . (8) p p2 p≤N

Finally, we observe that

∞ X 1 X 1 π2 < = − 1 < 1 p2 n2 6 p n=2

Proof of lemma 1.7: Put f(v) := (1 − v) exp(v + v2). We claim 1 ≤ f(v) for all v ∈ [0, 1/2]. To prove this, we check f(0) = 1 and

f0(v) = v(1 − 2v) exp(v + v2) which is nonnegative for v ∈ [0, 1/2]. 2 This completes Proof 2 of theorem 1.5. 2

We will prove later

X 1 1 = log log(N) + A + O , (9) p log(N) p≤N where A is a constant.

Question: Is it possible to prove (9) with O(1) in place of A + O(...) using only methods of Proof 2?

9 P 1 29.1.99 Third proof that p p diverges: The following identity holds for all σ > 1:

X 1 log(ζ(σ)) = − log 1 − (10) pσ p ∞ ∞ X X −1 X 1 X X 1 = − = + mpmσ pσ mpmσ p m=1 p p m=2

We will prove the identity (10) later. Note however, that the series involved converge absolutely. We claim: The last double sum on the right-hand side is bounded. Proof:

∞ ∞ X X 1 X X 1 < mpmσ pmσ p m=2 p m=2 X 1 1 X 1 = ≤ 2 ≤ 2ζ(2) p2σ 1 − 1 p2σ p p2σ p

1 1 because 1 − p2σ ≥ 2 . Note in passing:

Our bound holds for all σ ≥ 1, and the double sum converges for all σ > 1/2.

So we can summarise: X 1 log(ζ(σ)) = + O(1). pσ p

The left-hand side goes to inﬁnity as σ → 1, so the sum on the right-hand side must do the same. This completes proof 3 of theorem 1.5. 2

Proof of equation (10): Let P be a large prime. Then repeat the argument leading to 1 X 1 1 − ζ(σ) = 1 + 2σ nσ 2 < n odd

10 with all the primes 3, 5,...,P . This gives

1 1 1 1 X 1 1 − 1 − 1 − · ... · 1 − ζ(σ) = 1 + . 2σ 3σ 5σ P σ nσ p|n⇒p>P

The last sum ranges only over those n with all prime factors bigger than P . So it is a subsum of a tail end of the series for ζ(σ), hence tends to zero as P goes to inﬁnity. This gives

−1 Y 1 ζ(σ) = 1 − (11) pσ p and taking logs, we obtain (10). 2

Equation (11) is known as the ’Euler product representation of ζ’.

1.2 Euler’s Summation Formula This is a tool do derive sharp asymptotic formulae. Theorem 1.8 Given a real interval [a, b] with a < b, suppose f is a function on (a, b) with continuous derivative. Then X Z b Z b f(n) = f(t) dt + {t}f0(t) dt − f(b){b} + f(a){a}. (12) a

Here, {t} denotes the fractional part of t, deﬁned for any real t by

{t} = t − [t] and [t] is the greatest integer not bigger than t. Example:

{−2.1} = −2.1 − (−3) = 0.9.

Note that the formula applies to real as well as complex-valued functions.

Proof in the case a, b ∈ N: Suppose a < n − 1 < n ≤ b, then Z n [t]f0(t) dt = (n − 1)[f(n) − f(n − 1)] = nf(n) − (n − 1)f(n − 1) − f(n). n−1

11 Sum this from a + 1 to b:

b Z b X [t]f0(t) dt = (n − 1)[f(n) − f(n − 1)] a n=a+1 b X = bf(b) − af(a) − f(n). n=a+1

Rearrange this:

b X Z b f(n) = bf(b) − af(a) − [t]f0(t) dt (13) n=a+1 a

R b On the other hand, play with a f:

Z n Z b b 0 f(t) dt = [tf(t)]a − tf (t) dt (14) a a Now equations (13) and (14) together give the requested

b X Z b Z b f(n) = f(t) dt + {t}f0(t) dt. n=a+1 a a 2

PN 1 We apply the ESF to the harmonic series 2 k . Here, a = 1, b = N and f(t) = 1/t. Note that a = 0 would not work! ESF gives

N X 1 Z N 1 Z N {t} Z N {t} = dt − dt = log(N) − dt. (15) n t t2 t2 2 1 1 1 Now obviously

Z N {t} Z ∞ {t} Z ∞ {t} 2 dt = 2 dt − 2 dt 1 t 1 t N t

R ∞ 1 1 and the last term is less than N t2 dt = N . So we have proved (add One on both sides of (15))

12 Proposition 1.9

N X 1 1 = log(N) + γ + O , where n N n=1 Z ∞ {t} γ := 1 + 2 dt 1 t is known as the EULER-MASCHERONI-constant (some call it just EULER’s constant).

See the MTH Website for the ’constants’ page’, or see

http://www.mathsoft.com/asolve/constant/euler/euler.html

Deﬁnition 1.10 For 1 ≤ n ∈ N, let d(n) := number of divisors of n.

E. g. d(n) = 2 iﬀ n is a prime.

Information about d reﬂects something - crudely - about the distribution of the primes themselves.

Proposition 1.11

N X √ d(n) = N log(N) + (2γ − 1)N + O( N). (16) n=1

13 Proof: ESF in the usual form with integer boundaries just gives a remainder 1.2.99 O(N). For the sharper result, we have to apply ESF in the more general form as stated in theorem 1.8. But ﬁrst, look how the general form applies to the harmonic series: X 1 1 {x} = log(x) + γ + O + . n x x 1≤n≤x

1 The last summand is O( x ), so goes into the remainder term. The general form does not give us more information, and this is the case in most examples. But now we come to an example, where we really need it. We will use that √ #{(m, q): mq ≤ x} = 2#{(m, q): mq ≤ x, m < q} + O( x). (17) √ If mq ≤ x and m < q, then m < x. The number of q in this set for ﬁxed m is [x/m] − m (draw it!). X X X d(n) = 1 n≤x m≤x x q≤ m X √ = 2 1 + O( x) mq≤x m,q: m

Now we can attack each sum with ESF and obtain X √ x √ √ d(n) = 2x(log( x) + γ + O(x−1/2)) − 2 + O( x) + O( x) 2 n≤x √ = x log(x) + (2γ − 1)x + O( x) as stated. 2

Let us do another nice example: STIRLING’s formula says

log(N!) = N log(N) − N + O(log(N)). (18)

14 PN Proof: log(N!) = n=2 log(n). Put f(t) = log(t), so by ESF Z N Z N {t} log(N!) = log(t) dt + dt = N log(N) − N + O(log(N)). 1 1 t 2

1.3 Multiplicative arithmetical functions Deﬁnition 1.12 An arithmetical function is any function f : N → C, e. g. f(n) = 1/nσ or φ(n) = #{1 ≤ a ≤ n :(a, n) = 1}, (19) called the Euler phi-function. An arithmetical function such that f(1) 6= 0 and f(mn) = f(m)f(n) whenever m and n are coprime is called multiplicative (note that this implies f(1) = 1). If f has this property not only for coprime m, n, but for all m, n ∈ N, f is called completely multiplicative. Lemma 1.13 The Euler phi-function φ as deﬁned in (19) is multiplicative. Corollary 1.14 Q ep If n is factorized into powers of distinct primes, n = p p , then Y Y p − 1 φ(n) = (p − 1)pep−1 = n . p p|n p|n Exercise: Show that φ is not completely multiplicative but 1/nσ is. The proof of lemma 1.13 depends on the following result. Lemma 1.15 (Chinese Remainder Theorem - CRT) Suppose m, n ∈ N are coprime, then the simultaneous congruences x = a (mod m) x = b (mod n) have a solution x ∈ N for any a, b ∈ Z, and the solution is unique modulo mn.

15 The Chinese Remainder Theorem was discovered by Chinese mathematicians in the 4th century A.C. Proof of CRT: We claim that there exist m0, n0 such that mm0 = 1 (mod n)(∗) and nn0 = 1 (mod m). Put x := bmm0 +ann0, this satisﬁes both the required congruences. If, on the other hand, x and y satisfy both congruences, x − y is divisible by m and by n. Since m and n are coprime, x − y must be divisible by mn. Proof of claim (∗): First, we reformulate the claim:

a is coprime to n ⇔ a has multiplicative inverse modulo n ⇔ ∃b : ab = 1 (mod n). (20)

The implication from right to left is clear, only from left to right we have to work a bit. b is found using the Euclidean algorithm. We do an example that should make the method clear: E. g. to solve 11 · b = 1 (mod 17) we do

17 = 11 · 1 + 6 11 = 6 · 1 + 5 6 = 5 · 1 + 1.

The last non-zero remainder is the gcd, which is One by hypothesis. From this, work your way backwards, always expressing the remainder by the previous two terms:

1 = 6 − 5 = 6 − (11 − 6) = 2 · 6 − 11 = 2(17 − 11) − 11 = 2 · 17 − 3 · 11.

Example for CRT: Solve x = 2 (mod 17) and x = 8 (mod 11). We ﬁnd m0 = 2 and n0 = 14 congruent to −3 as in the proof of CRT. Then

x = 8 · (17 · 2) + 2 · (11 · 14) = 580 satisﬁes the two congruences (the smallest solution is the remainder of 580 divided by (11 · 17), namely 19).

Proof of lemma 1.13: Deﬁne a map

Φ: Z/(mn) → Z/m × Z/n (21)

16 simply by x 7→ (x mod m, x mod n). By CRT, Φ is a bijection (in fact, Φ is an isomorphism of rings). Now deﬁne U Z/m := {1 ≤ a ≤ m :(a, m) = 1} and likewise for n and mn. Since x is coprime to mn iﬀ it is coprime both to m and n, we can restrict Φ to the U-level: Φ: U Z/(mn) → U Z/m × U Z/n (22)

Here Φ is still a bijection (in fact, the sets U(...) are the units of Z/(mn) resp. Z/m resp. Z/n by (20), and Φ is an isomorphism of groups). By definition, the cardinality of U(Z/m) is just φ(m) and likewise for n and mn, which completes the proof of lemma 1.13. 2

Remark 1.16 CRT works in greater generality: If m1, . . . mr are pairwise coprime then for 2.2.99 any a1, . . . ar the simultaneous congruences x = ai (mod mi) have a solution x ∈ Z, and it is unique modulo m1 · ... · mr. Theorem 1.17 For any n ∈ N holds X φ(d) = n. d|n Proof: Check the equality for n = pr (a prime power). The left hand side is

r−1 X 1 + (p − 1)pi−1 = 1 + (pr − 1) = n i=0 as a sum of a geometric progression. Next, observe that both sides of the equation in theorem 1.17 are multiplicative arithmetic functions. For the left-hand side, this follows from X X X X X φ(d) = φ(d1d2) = φ(d1) φ(d2)

d|mn d1|m d2|n d1|m d2|n for any pair of coprime integers (m, n) (note that d divides mn iﬀ there exist divisors d1 of m and d2 of n such that d = d1d2). So it was enough to check the prime power case. 2

17 Definition 1.18 The Möbiusfunction µ is defined by

µ(1) = 1 µ(n) = (−1)k if n is a product of k distinct primes µ(n) = 0 otherwise.

This course is built around the strange phenomenon, that in order to study integers, especially primes, one needs to study functions. We will prove after Easter (see Apostol, p. 91): X PNT ⇐⇒ µ(n) = o(x), i. e. n≤x 1 X µ(n) → 0. (23) x n≤x

Also, the Riemann hypothesis is equivalent to a result about partial sums of µ.

Theorem 1.19 The M¨obiusfunction µ is multiplicative. Furthermore

X 1 if n = 1 µ(d) = 0 otherwise. d|n

Proof: First, we have to show µ(mn) = µ(m)µ(n) for coprime integers (m, n). Factorize m and n as product of prime powers. The primes involved must all be distinct. If anywhere in the factorisation is an exponent of at least Two, we obviously get an equation 0 = 0. If m and n are products of k resp. l distinct primes, then mn is a product of k + l primes, and they are all distinct, since m and n are coprime. So we get µ(m) = (−1)k, µ(n) = (−1)l and µ(mn) = (−1)k+l = µ(m)µ(n). For the next claim, it is suﬃcient to check the prime power case, since again both sides of the equation of theorem 1.19 are multiplicative (the same argument as in the proof of theorem 1.17). If n = pr with r ≥ 1, the left-hand side is µ(1) + µ(p) = 1 − 1 = 0, and this is all we have to do! 2

18 Theorem 1.20 X n φ(n) = µ(d) (24) d d|n Proof: We know from corollary 1.14 φ(n) = n Q 1 − 1 , so pi|n pi

φ(n) X 1 X 1 X µ(d) = 1 − + − ... = . n pi pipj d pi|n i6=j d|n 2

19 1.4 Dirichlet Convolution 5.2.99 Theorem 1.20 is a special instance of a general technique: Deﬁnition 1.21 For arithmetical functions f, g let X n (f ∗ g)(n) := f(d)g . d d|n

This is a new arithmetical function, called the convolution of f and g.

Theorem 1.22 Convolution is commutative and associative. In symbols: For all arithmetical functions f,g,h holds i) f ∗ g = g ∗ f and ii) (f ∗ g) ∗ h = f ∗ (g ∗ h).

Proof of i): The sum in X n f(d)g d d|n runs over all pairs d, e ∈ N with de = n, so it is equal to X f(d)g(e), de=n and the latter expression is clearly symmetric (f and g can be interchanged). Proof of ii): Do for example n = p by hand! the proof in general goes much in the same way as the proof of i): Both sides of ii) are equal to X f(c)g(d)h(e). cde=n 2

Lemma 1.23 Deﬁne the arithmetical function I by I(1) = 1 and I(n) = 0 for all n > 1. Then for all arithmetical functions f,

f ∗ I = I ∗ f = f.

20 Proof: By deﬁnition, X n f ∗ I(n) = f(d)I = f(n)I(1) = f(n) d d|n

(all the other summands are zero by the deﬁnition of I). 2

Theorem 1.24 If f is an arithmetical function with f(1) 6= 0, there exists exactly one arithmetical function g such that f ∗ g = I. This function is denoted by f−1.

Proof: The equation (f ∗ g)(1) = f(1)g(1) determines g(1). Then deﬁne g recursively: Assuming that g(1),..., g(n − 1) has been deﬁned (and that there is only one choice), the equation X n f ∗ g(n) = f(1)g(n) + f(d)g d 1

Example: Let u(n) = 1 for all n. Then we have, by theorem 1.19

u−1 = µ. (25)

Theorem 1.25 (M¨obiusInversion Formula) Given arithmetical functions f and g, the following are equivalent: X X n f(n) = g(d) ⇐⇒ g(n) = f(d)µ . d d|n d|n

Proof of ⇒: Let u(n) = 1 for all n as in the example. Then convolve both sides of f = g ∗ u with µ and use (25)

f ∗ µ = g ∗ u ∗ µ = g ∗ I = g.

This proves the implication from left to right. For the converse, convolve with u. 2

21 Corollary 1.26 Theorems 1.17 and 1.20 are equivalent, e. g. from theorem 1.17 we obtain a proof of theorem 1.20 by convolving with the M¨obiusfunction.

Corollary 1.27 (Application) Suppose we have absolutely convergent series of the form

X f(n) X g(n) F(σ) = , G(σ) = . nσ nσ N N Then X (f ∗ g(n)) F(σ) · G(σ) = . nσ N Example: If f ∗ g = I, we get F(σ)G(σ) = 1. So

1 X µ(n) = . ζ(σ) nσ N Series as those for F, G and F · G are called Dirichlet series. We are now going to study the Riemann zeta function in the context of Dirichlet series. 8.2.99 Add to Q14: For all s such that <(s) > 2,

∞ ζ(s − 1) X φ(n) = . ζ(s) ns n=1

22 2 The Riemann Zeta Function

The traditional notation for the variable is

s = σ + it with σ, t ∈ R. Claim: For all s such that <(s) > 1, the series P 1 converges absolutely. N ns Proof of claim: Calculate the modulus of each term: n−s = n−σ−it = n−σe−it log(n) has modulus n−σ, and we know already that P 1 is a convergent series. N nσ

2.1 Euler Products We recall equation (11):

−1 Y 1 ζ(σ) = 1 − . pσ p We will show that this holds for all complex s with <(s) > 1. Since it is not more diﬃcult, we prove a more general theorem:

Theorem 2.1 If f is a multiplicative arithmetical function, and P f(n) converges abso- N lutely, then X Y f(n) = (f(1) + f(p) + f(p2) + ...) (26) N p and if moreover, f is completely multiplicative, then X Y 1 f(n) = . 1 − f(p) N p Proof of theorem 2.1: Let Y P (x) := (f(1) + f(p) + f(p2) + ...). p≤x Here, each factor is absolutely convergent, and we have a ﬁnite number of factors, so X P (x) = f(n) n∈A

23 where A := {n ∈ N : all prime factors of n are less than x}. Now estimate the diﬀerence

X X X X f(n) − f(n) ≤ |f(n)| ≤ |f(n)|. N A N−A n>x The last sum tends to zero as x tends to inﬁnity, because it is the tail end of a convergent series. The identity for completely multiplicative functions follows from the general Euler product expansion, because in this case each factor of the inﬁnite product is a convergent geometric series. 2

Remark 2.2 An infinite product is defined to be convergent, if the corresponding partial products form a convergent sequence, which does NOT converge to zero. The non-zero condition is imposed, because we want to take logs of infinite products. Note that it is automatically satisfied in the setting of theorem 2.1 for all completely multiplicative functions f (e. g. f(n) = n−s): The limit of a convergent geometric series cannot be zero.

Deﬁnition 2.3 Just a quick reminder: If S is an open subset of C, a function f : S → C is called complex diﬀerentiable or holomorphic on S if the limit

f(z + h) − f(z) lim h→0 h exists and is ﬁnite for all z ∈ S. If for all z ∈ S, f equals its own Taylor series in a small neighbourhood of z,

∞ X f(n)(z) f(z + h) = hn, n! n=0 f is called analytic on S. We know from complex analysis that all functions holomorphic on S are analytic on S and vice versa (whereas for real functions, ’analytic’ is a stronger condition than ’diﬀerentiable inﬁnitely often’).

24 Our next goal is that ζ(s) is analytic in the half-plane <(s) > 1. Non-proof: Each n−s is analytic, so the sum is analytic with derivative P − log(n) . Unfortunately, you can’t guarantee that an inﬁnite sum of ana- N ns lytic functions is analytic. Warning example: Put for all n ≥ 0 x2 f (x) = . n (1 + x2)n

We can sum the fn, because we can sum a geometric progression.

N X x2 x2 fn(x) = − 1 1 N+1 n=0 1 − 1 − 1 + x2 1 + x2 1 + x2 Now if x 6= 0 we can let N tend to inﬁnity, the second term vanishes, and 2 the whole sum converges to f(x) = 1 + x . But for x = 0, all fn(0) = 0, so the limit f(0) = limn fn(x) = 0, too. So the limit function f is not even continuous, although all the fn are analytic in a neighbourhood of 0! If you look at the complex analogue, you get the same phenomenon in the region {s : | arg(s)| < π/4}. One useful way to make sure that nothing goes wrong is the concept of uniform convergence.

2.2 Uniform Convergence Deﬁnition 2.4 Given a non-empty subset S of C, a function F and a sequence (FN ) of functions on S, we say that FN (s) converges to F(s) if for all ε > 0 there exists N0 = N0(ε, s) such that for all N > N0,

|F(s) − FN (s)| < ε.

We say that the sequence FN converges to F uniformly on S, if for all ε > 0 there exists N0 = N0(ε) such that for all N > N0 and for all s ∈ S

|F(s) − FN (s)| < ε.

Warning: the N0 in the deﬁnition of uniform convergence must not depend on s. Lots of nice properties of limits of uniformly convergent sequences of functions are inherited. The ﬁrst example for this is

25 Proposition 2.5 Suppose that the sequence of functions (FN ) converges to F uniformly on S. If all FN are continuous on S, then F is continuous on S.

Proof: Let s0 ∈ S. Given ε > 0, choose N such that for all s ∈ S, ε |F(s) − F (s)| < . (27) N 3

This is possible since the FN are converging uniformly on S. Next, choose δ > 0 such that for all s ∈ S with |s − s0| < δ, ε |F (s) − F (s )| < . (28) N N 0 3

Now we have set the stage: For all s ∈ S with |s − s0| < δ, we have

|F(s) − F(s0)| = |F(s) − FN (s) + FN (s) − FN (s0) + FN (s0) − F(s0)|

≤ |F(s) − FN (s)| + |FN (s) − FN (s0)| + |FN (s0) − F(s0)| ε ε ε < + + = ε. 3 3 3 Here, we have used (27) twice, in order to estimate the ﬁrst and third term, and (28) for the second term. This proves that F is continuous in an arbitrary s0 ∈ S. 2

26 Proposition 2.6 For every δ > 0, the partial sums of the Riemann zeta function converge 9.2.99 uniformly on S1+δ := {s ∈ C : <(s) > 1 + δ}. That is, N X 1 → ζ(s) ns n=1 uniformly on S1+δ. Hence, by proposition 2.5 the Riemann zeta function is continuous on [ S1+δ = {s ∈ C : <(s) > 1} = S1. δ>0

Note that the convergence is not uniform on the whole of S1. Proof of proposition 2.6:

N ∞ ∞ X 1 X 1 X 1 ζ(s) − = ≤ ns ns nσ n=1 n=N+1 n=N+1 Now we need to show that this is less than ε in a way independent of σ. Use ESF! ∞ X 1 Z ∞ 1 Z ∞ 1 0 = dt + {t} dt. (29) nσ tσ tσ n=N+1 N N The ﬁrst term in the right-hand side of equation (29) is t1−σ ∞ −N 1−σ N 1−σ = ≤ 1 − σ N 1 − σ δ 1 ≤ < ε (30) δN δ for all N large enough, i. e. N > N0(δ, ε). Note that the inequality (30) depends only on δ, not on σ. The second term can be estimated by Z ∞ 1 0 1 1 σ dt = σ ≤ 1+δ < ε, N t N N as soon as N > N0(δ, ε) with the same N0 as above (in fact, the second term is smaller than the ﬁrst term). Hence for all N greater than N0(δ, ε/2), the series in equation (29) is less than ε. This completes the proof of proposition 2.6. 2

27 2.3 The Zeta Function is Analytic Theorem 2.7 Suppose S ⊆ C is open, and we have a function F : S → C and a sequence of functions FN : S → C converging to F uniformly on S. If all the FN are analytic, then F is analytic.

PN 1 Example: FN (s) = 1 ns is analytic and converges uniformly to ζ(s) on every S1+δ, δ > 0 as in proposition 2.6, so by theorem 2.7, the Riemann zeta function is analytic in S1. Proof of theorem 2.7: Given a fixed point a ∈ S, we have to prove that F is analytic in a neighbourhood of a. We want to use complex analysis, in particular Cauchy’s formula: Let γ be a closed simple curve, which is a finite join of smooth curves, such that a ∈ Int(γ) and the closure Int(γ) ⊆ S. Then Cauchy’s formula says that for any function f which is analytic in S, and for any b ∈ Int(γ) 1 Z f(z) f(b) = dz. (31) 2πi γ z − b Since S is open, it contains a small disc around a. So we can choose γ as a sufficiently small circle around a contained in S, but our argument will work for any γ with the above properties. We will need to use the following

Lemma 2.8 Suppose a sequence of continuous functions GN : γ → C converges to a function G : γ → C uniformly on γ. Then G is continuous and we have Z Z lim GN (s) ds = G(s) ds. N→∞ γ γ Proof of lemma 2.8: The continuity of G follows from proposition 2.5, so in particular G is integrable. Now Z Z Z

G(s) ds − GN (s) ds = G(s) − GN (s) ds γ γ γ Z ≤ |G(s) − GN (s)| ds γ

≤ length(γ) max |G(s) − GN (s)|. s∈γ

28 The last quantity tends to zero by the deﬁnition of uniform convergence. This completes the proof of the lemma. 2

Now the magic wand is waved! By hypothesis, the functions FN are analytic in S. So for all b ∈ Int(γ) ⊆ S by Cauchy’s formula (31) Z 1 FN (s) FN (b) = ds 2πi γ s − b

FN (s) F(s) Put GN (s) := s−b . Then we claim that GN converge to G(s) := s−b uniformly on γ:

F(s) − FN (s) |G(s) − GN(s)| = s − b

≤ C max |F(s) − FN (s)| s∈γ 1 where C := maxs∈γ |s−b| . This proves that for all b ∈ Int(γ) 1 Z 1 Z F(s) F(b) = lim FN (b) = lim GN (s) ds = ds. (32) N N 2πi γ 2πi γ s − b Here, we have applied lemma 2.8 to interchange limit and integral in the last step. Finally, recall that any function F satisfying equation (32) (Cauchy’s formula) in Int(γ) is analytic there. A proof of this step: For all b ∈ Int(γ), F(b + h) − F(b) 1 Z 1 1 = F(s) − ds h 2πih γ s − b − h s − b 1 Z F(s) = ds, 2πi γ (s − b)(s − b − h) the h’s cancel! In the last integral, the limit h → 0 may be taken and gives the derivative. Strictly speaking, we would have to check for uniform convergence (as with the GN above) to be able to apply lemma 2.8 again. 2

Corollary 2.9 For all s with <(s) > 1, d X log(n) ζ(s) = . ds ns N Problem: the real action takes place to the left of <(s) = 1. But our deﬁnition of the zeta function does not work there!

29 2.4 Analytic continuation of the Zeta Function Further reading on this topic:

Titchmarsh: . . . Riemann zeta function (updated by Roger Heath-Brown).

Apostol is a little bit ponderous on this - he does the most general zeta function that he can.

An easy example: Consider the function

g(s) = 1 + s + s2 + ...

The series converges for |s| < 1. Claim: g can be continued to a function which is analytic on the whole of C except for a simple pole at s = 1. Proof: −1 For |s| < 1, g(s) = s−1 . The latter expression is deﬁned on C apart from a simple pole at s = 1 with residue −1. BUT: g is not deﬁned by the series for |s| ≥ 1. 12.2.99

Theorem 2.10 The Riemann zeta function is analytic on the whole of the complex plane with exception of a simple pole at s = 1, with residue 1.

Proof: In this section, we will only prove that there is an analytic continuation to the half-plane <(s) > 0. For this, we will see three methods of proof (theorem 2.10 will then be an immediate consequence of the functional equation which we will prove in chapter 3). The three methods are

1. Standard - the one you will ﬁnd in the books.

2. A very elegant one, which you will do in exercise 16.

3. One which I do not claim to have invented - I rather think, it must be known to the experts. But I did not ﬁnd it in the books.

1. The standard method: Use the ESF. But be careful: ESF is about ﬁnite intervals!

N X 1 Z N Z N −s{t} = t−s dt + dt (33) ns ts+1 n=2 1 1

30 The first term is t1−s N N 1−s 1 = − , 1 − s 1 1 − s 1 − s and since we suppose <(s) > 1, we get N 1−s → 0 as N tends to infinity. The second term in equation (33) converges too, as N tends to infinity, since

Z ∞ |s| σ+1 dt < ∞ 1 t gives an upper bound. So we are justiﬁed in writing

∞ X 1 1 Z ∞ {t} ζ(s) = 1 + = 1 − − s dt. (34) ns 1 − s t1+s n=2 1 We claim: the integral in equation (34) represents an analytic function in the range <(s) > 0. By this claim, we get the analytic continuation of ζ to the half-plane <(s) > 0. There it is analytic, apart from a simple pole at s = 1 with residue 1. Proof of claim: write ∞ X I(s) = fn(s), (35) n=1 R n+1 {t} where fn(s) := n ts+1 dt. We will prove that for any δ > 0, a) the series for I(s) converges uniformly on <(s) > δ and

b) each fn is analytic in the range <(s) > δ.

Then we may apply theorem 2.7 to complete the proof that I(s) is analytic on <(s) > 0. As to a),

N ∞ ∞ X X X I(s) − fn(s) = fn(s) ≤ |fn(s)| n=1 n=N+1 N+1 Z ∞ 1 t−σ ∞ ≤ σ+1 dt = N+1 t −σ N+1 (N + 1)−σ = , σ

31 1 and in absolute value, this is smaller than δ(N+1)δ , so it tends to zero. And the bound depends on δ only, not on σ or s! This proves uniform convergence. As to claim b), consider the diﬀerence quotient

Z n+1 fn(s + h) − fn(s) 1 {t} 1 = s+1 h − 1 dt. (36) h h n t t Use a ﬁrst-order Taylor approximation of the exponential:

t−h = e−h log(t) = 1 − h log(t) + f(h, t), where f(h, t) = O((h log(t))2). (37)

Put this into equation (36):

1 Z n+1 {t} 1 (fn(s + h) − fn(s)) = s+1 − log(t) + f(h, t) dt. (38) h n t h We can guess the derivative:

Z n+1 Z n+1 1 {t} 1 (fn(s + h) − fn(s)) + s+1 log(t) dt ≤ σ+1 |f(h, t)| dt. (39) h n t n ht The right-hand side tends to zero as h → 0 by (37). This completes the ﬁrst proof for the analytic continuation of the zeta function into <(s) > 0. 2

3. GE’s method: Has the additional beneﬁt of giving a continuation to the whole of the complex plane (with exception of the simple pole at s = 1). Consider Z ∞ −1 1 x−s dx = = . 1 1 − s s − 1 Subtract this from zeta, to remove the pole! For <(s) > 1,

∞ ∞ 1 X 1 X Z n+1 ζ(s) − = − x−s dx s − 1 ns n=1 n=1 n Z 1 X 1 −s = s − (n + x) dx n 0 N X 1 Z 1 x−s = s 1 − 1 + dx . (40) n 0 n N

32 The re-arrangement is ok, because the sums involved converge absolutely in <(s) > 1. Now assume that we have continued the zeta function to the domain <(s) > 2−K, for some integer K ≥ 0. We want to continue it further to <(s) > 1 − K. To do this, put h := x/n and use a Taylor approximation of −s fs(h) := (1 + h) of order K. Recall that the Taylor polynomial of degree K for fs in h = 0 is deﬁned by K (k) X fs (0) T (h) := hk. (41) f,s,K k! k=0 (If you only want the continuation to <(s) > 0, think of K = 1: The Taylor −s sx polynomial for (1 + x/n) in this case is simply 1 − n , the error term is O(n−2). Put this into equation (40), and you get a series convergent for <(s) > 0). We have to calculate higher derivatives of fs in h = 0. They are given by equating h = 0 in (−1)k(s + k − 1)(s + k − 2) · ... · (s + 1)s f(k)(h) = f (h) (42) s (h + 1)k s

(it is easy to prove the relation (42) by induction on k). Since fs is analytic in the neighbourhood of h = 0, we have an estimate for the error term:

|f(K+1)(h0)| |f (h) − T (h)| ≤ s |h|K+1 (43) s f,s,K (K + 1)! for some h0 with |h0| < |h|. For bounded values of s, this is an O(|h|K+1). Use the Taylor polynomial with h = x/n in equation (40). We evaluate the inner integral ﬁrst:

Z 1 K (k) x−s X fs (0) 1 1 + dx = 1 + + O . (44) n (k + 1)! nk nK+1 0 k=1 Putting this into equation (40) gives the nice identity

K 1 X (−1)k(s + k − 1) · ... · (s + 1)s ζ(s) − = − ζ(s + k) s − 1 (k + 1)! k=1 ∞ X 1 Z 1 x x−s + T − 1 + dx. (45) ns f,s,K n n n=1 0

33 The last sum converges by (43), and for all s 6= 0, −1, −2,..., 2 − K the values ζ(s+1), ζ(s+2), . . . , ζ(s+K −1) are all deﬁned by hypothesis, giving the continuation of the zeta function to <(s) > 1 − K. In case s = −m = 0, −1, −2,..., 2 − K, one of the arguments of ζ in (45) becomes 1, but this is no problem, since the pole is cancelled out by the appropriate factor (s + m) in the coeﬃcient (the right-hand side of (45) has a removable singularity there). Thus, we get an analytic continuation of the zeta function to

<(s) > −1, −2,..., in fact, to the whole of the complex plane! But in the next section, we will see a better method to achieve this: via the functional equation. 2

Too late for beginning with the functional equation - here’s a nice formula!

Theorem 2.11 If s = σ + it, σ > 1, then

Z ∞ π(x) log(ζ(s)) = s s dx. 2 x(x − 1) Proof: See exercise 29, hint: You will obtain a sum over all primes. Convert this to a sum over all natural numbers using

1 if n is prime, π(n) − π(n − 1) = 0 otherwise.

In Q19 (MAPLE): Change mod to abs. Q 7,8,9,10,13,14 in by Fri wk 5.

34 3 The Functional Equation 15.2.99 3.1 The Gamma Function It is amazing how the Gamma function helps us to understand the zeta function. Deﬁnition 3.1 For every s with <(s) > 0, the integral Z ∞ Γ(s) := e−tts−1 dt 0 exists and is called the Gamma function Γ(s). The integral exists in this range by comparison with the real Gamma function, see MTH2A32. We claim: Γ is analytic in <(s) > 0. The proof is left as an exercise - proceed along the following steps:

P∞ R n+1 −t s−1 I : Γ(s) = 0 Γn(s), where Γn(s) := n e t dt, PN II : 0 Γn(s) → Γ(s) uniformly on <(s) > δ, for any ﬁxed δ > 0, III : Prove that Γn is analytic.

R ∞ {t} All these steps are very similar to the argument for 0 ts+1 dt. Later on, another (better) proof will be given.

Proposition 3.2 The Gamma function has the following properties: 1. For all s with <(s) > 0,

Γ(s + 1) = sΓ(s).

2. For all integers N ≥ 0,

Γ(N + 1) = N!.

Proof of 1: Z ∞ Z ∞ −t s −t s ∞ −t s−1 Γ(s + 1) = e t dt = [−e t ]0 + s e t dt. 0 0

35 The bracket term vanishes at t = 0 because <(s) > 0. Item 2. follows from 1. by induction together with Γ(1) = 1. 2

Now write 1 Γ(s) = Γ(s + 1). (46) s The right-hand side is deﬁned for <(s) > −1 apart from s = 0, where it has a simple pole with residue Γ(1) = 1. Iterate this: 1 Γ(s) = Γ(s + 2). (47) s(s + 1) The right-hand side of (47) is deﬁned for <(s) > −2, apart from s = 0, −1, where we have simple poles again. In this way, we can continue the Gamma function to the whole plane, where it is analytic apart from simple poles at 0, −1, −2,.... Our goal throughout this chapter will be the following theorem:

Theorem 3.3 (THE Functional Equation) Deﬁne s F(s) := π−s/2Γ ζ(s), 2 known to exist for <(s) > 0. Then F satisﬁes the functional equation F(1 − s) = F(s).

Corollary 3.4 1. F is analytic in the whole complex plane apart from poles at 1 and 0. 2. If we know that Γ never vanishes, then ζ can be continued to the whole plane, where it is analytic apart from a simple pole at s = 1.

For we can expand theorem 3.3, giving − 1−s 1 − s − s s π 2 Γ ζ(1 − s) = π 2 Γ ζ(s), 2 2 1 s −s+ 2 π Γ 2 ζ(s) ζ(1 − s) = 1−s . (48) Γ 2 Example: We know that Γ has a simple pole at −m for m ∈ N. So, for all m ∈ N, ζ(−2m) = 0 (49)

36 (we have 1 − s = −2m ⇐⇒ s = 2m + 1, we know ζ 6= 0 for <(s) > 1 by the Euler product expansion and we assumed Γ 6= 0 everywhere). The case 1 − s = 0, i. e. s = 1 is diﬀerent: There, the right-hand side has a simple pole in the numerator too (in ζ), cancelling the one in Γ. Thus ζ is analytic and nonzero in s = 0. Picture: By the functional equation, all we need to know is the values of ζ for <(s) ≥ 1/2. We found ζ(−2m) = 0 for all m ∈ N. There are no more zeros of ζ with <(s) < 0, because Γ has no other poles (just look at equation (48)). Also, ζ(s) 6= 0 for <(s) > 1 because of the Euler product expansion (see remark 2.2). So any other zero of ζ must lie in the ’critical strip’ 0 ≤ <(s) ≤ 1. Riemann stated without proof that all zeros of ζ in the critical strip have <(s) = 1/2 (the ’Riemann Hypothesis’). This has not been proved yet. A very important unsolved problem - see the talk by Michael Berry, FRS. All the zeros found thus far lie on the line <(s) = 1/2, and they are all simple.

3.2 Fourier Analysis For the proof of the functional equation - theorem 3.3 - we will need some Fourier analysis. Definition 3.5 The Schwartz space S is the set of functions f : R → C which are infinitely differentiable, and whose derivatives f(n) (including f(0) = f) all satisfy (1 + |x|)m f(n) = O(1) (50)

2 for all m ∈ N. The bound may depend upon m and n. Example: e−x ∈ S. Note that S is a complex vector space, and that all functions f ∈ S are integrable: Z Z Z 1 f(x) dx ≤ |f(x)| dx ≤ C 2 dx R R R (1 + |x|) (take n = 0 and m = 2 in equation (50)). Now define for all f ∈ S the Fourier transform of f by Z ˆf(y) := f(x) exp(−2πixy) dx. R 37 This integral exists for the same reason as above: Z |ˆf(y)| ≤ |f| < ∞, R in fact ˆf ∈ S again (apply equation (50) with m + n to get the bound for ˆf(n)). Thus f → ˆf is a linear map from S to S. Periodicity: A function g on R is periodic if g(x) = g(x + m) for all m ∈ Z and for all x ∈ R. Suppose g is periodic and piecewise continuous. Then its k-th Fourier coefficient is defined for k ∈ Z as Z 1 −2πikx ck := g(x)e dx. 0 Lemma 3.6 If g is periodic and differentiable infinitely often, then there exists C > 0, 16.2.99 depending only upon g, such that C |c | ≤ k k2 for all k 6= 0.

Proof: Integrate by parts!

−e−2πikxg(x)1 Z 1 −e−2πikxg0(x) ck = + dx. 2πik 0 0 2πik Now the bracket term vanishes because g is periodic. Integrate by parts again, so that k2 appears in the denominator, then estimate the exponential R 1 00 2 by 1. Put C = 0 |g |dx/(4π ). 2

Theorem 3.7 Any function g which is periodic and diﬀerentiable inﬁnitely often has a Fourier series expansion X 2πikx g(x) = cke k∈Z which is uniformly convergent.

38 Proof that the Fourier series converges uniformly: Call G the Fourier series of g, and apply lemma 3.6.

n X X 1 G(x) − c e2πikx ≤ C , k k2 k=−n |k|>n where the last sum tends to zero independent of x (the constant C depends only on g). The equality g(x) = G(x) for all x ∈ [0, 1] is not so easy to prove. Since this is not really part of the course, the proof is done in Appendix A. 2

Theorem 3.8 (Poisson Summation Formula - PSF) Suppose f belongs to the Schwartz space (see definition 3.5. Then X X f(m) = ˆf(m). Z Z Proof: Let X g(x) := f(x + m). Z Clearly g is periodic. But g is also differentiable infinitely often:

X (n) X (n) X 1 f (m) ≤ f (m) ≤ C , (1 + |x + m|)2 |m|>N |m|>N |m|>N where the last series tends to zero independent of x. So the n-th derivatives of the partial sums converge uniformly, for all n ≥ 0. Now we cannot use theorem 2.7, since the fn are not necessarily analytic. But we can use lemma 2.8: Let γ be the real interval [1, x], let the functions GN be the partial sums 0 of the derivatives fn and use the fundamental theorem of calculus: d Z x GN (t) dt = GN (x) − GN (0). dx 0 Here, the integral converges to g(x) − g(0) as N → ∞. (and similarly for higher derivatives, using induction). Therefore g is n times diﬀerentiable and its n-th derivative is the limit of that of the partial sums. So we may do Fourier analysis!

39 Let ck be the k-th Fourier coeﬃcient of g, so by theorem 3.7

X 2πikx X g(x) = cke , g(0) = ck. (51) k∈Z k∈Z On the other hand, Z 1 Z 1 −2πikx X −2πikx ck = g(x)e dx = f(x + m)e dx 0 0 m∈Z X Z 1 = f(x + m)e−2πikx dx. 0 m∈Z This interchange of sum and integral is justiﬁed, because the series for g converges uniformly (lemma 2.8 again). We multiply each summand by a factor of e−2πikm = 1: Z 1 Z X −2πik(x+m) −2πikx ck = f(x + m)e dx = f(x)e dx 0 m∈Z R = ˆf(k). Now use equation (51): X X Xˆ f(m) = g(0) = ck = f(k). m∈Z k∈Z k∈Z This completes the proof of the Poisson summation formula. 2

Lemma 3.9 If f(y) = e−πy2 , then ˆf(y) = f(y).

Proof: Z ˆf(y) = e−πy2 e−2πikxy dx. R The idea is to complete the square: −π(x2 + 2ixy) = −π[(x + iy)2 + y2]. So the Fourier transform of f is Z ˆf(y) = e−πy2 e−π(x+iy)2 dx. R 40 Call I(y) the integral R exp(−π(x + iy)2) dx. We know that I(0) is One. R What happens if y 6= 0? Fix some large N and consider the following paths:

γ1 := [−N,N], γ2 := [N,N + yi], γ3 := [N + yi, −N + yi], γ4 := [−N + yi, −N].

−πz2 Put γ := γ1 + γ2 + γ3 + γ4 (a rectangle). Since e is an analytic function in the whole of the complex plane, we have for any N Z e−πz2 dz = 0. γ

−πz2 Now as N → ∞, the integral of e over γ1 tends to I(0) = 1, the integral over γ3 tends to −I(y) and the integrals over γ2 and γ4 both tend to 0, by the standard estimate. This completes the proof of lemma 3.9. 2

Could we make up an argument involving only real integrals? Sure, we could have sin and cos and real part etc etc. But I do not see at all how this could work - certainly some clever tricks would be necessary. No chance that it would be more elegant than complex integration.

3.3 The Theta Function Theorem 3.10 Deﬁne for real y > 0 the theta function by

X 2 θ(y) := e−n πy. Z Then 1 √ θ = y θ(y). (52) y Proof: This is not obvious! Does not even look possible. But it really makes the functional equation for the zeta function work. Note that the series defining θ converges uniformly in the range y > δ for any fixed δ > 0 (see also exercise 30). Fix some real b > 0 and define with our old friend f of lemma 3.9

−πb2y2 fb(y) := f(by) = e .

41 Of course fb is in the Schwartz space, so we may apply the Poisson summation formula 3.8 X Xˆ fb(n) = fb(n). (53) Z Z ˆ What is fb(y)? Z Z ˆ −2πixy −2πixy fb(y) = fb(x)e dx = f(bx)e dx. (54) R R

1 Now put u := bx, so dx = b du: Equation (54) becomes Z ˆ 1 −2πiu y 1ˆy fb(y) = f(u)e b du = f . (55) b R b b We apply lemma 3.9 to this equation, so 1 y ˆf (y) = f . (56) b b b Put this result into equation (53) and insert the deﬁnition of f again:

2 X −πb2m2 1 X −π m e = e b2 . b Z Z √ Finally, put b := y and the functional equation for θ emerges. 2

42 Now we are ready for the proof of theorem 3.3! We begin with 19.2.99 Z ∞ Z ∞ s −x s −1 −x s dx Γ = e x 2 dx = e x 2 . (57) 2 0 0 x So, in the domain <(s) > 1 + δ, Z ∞ − s X −s −x s dx F(s) = π 2 n e x 2 . (58) 0 x n∈N Next, replace x by πn2y in the integral. This means dx/x = dy/y, and we get after some cancelling Z ∞ X −πn2y s dy F(s) = e y 2 . (59) 0 y n∈N The interchange of integral and sum is ok, because the series for the zeta function converges uniformly on <(s) > 1 + δ. Deﬁne

X 2 θ(y) − 1 g(y) := e−πn y = . (60) 2 n∈N Split the integral in equation (59):

∞ 1 Z s dy Z s dy F(s) = y 2 g(y) + y 2 g(y) . (61) 1 y 0 y In the second integral, change y to z = y−1. Thus it becomes an integral from ∞ to 1, and dz/z = −dy/y (this minus sign goes into reversing the boundaries): Z ∞ Z ∞ s dy − s −1 dz F(s) = y 2 g(y) + z 2 g(z ) . (62) 1 y 1 z Now our preparations come into play: By theorem 3.10, which we proved using the Poisson summation formula, theorem 3.8 √ θ(y−1) − 1 y θ(y) − 1 g(y−1) = = √ 2 √ 2 √ y (θ(y) − 1) + y − 1 √ y − 1 = = y g(y) + . (63) 2 2 43 Put this into equation (62)

∞ ∞ Z s dy Z 1−s dy F(s) = y 2 g(y) + y 2 g(y) 1 y 1 y Z ∞ 1 − s 1 dy + y 2 y 2 − 1 . (64) 2 1 y

We evaluate the third integral I3 in equation (64):

∞ ∞ " 1−s − s # Z 1+s 2+s y 2 y 2 − 2 − 2 2I3 = y − y dy = 1−s − s 1 − 2 2 1 −1 1 1 1 = 2 − = 2 − . (65) 1 − s s s − 1 s

Claim: For all z ∈ C, the function Z ∞ G(z) = yzg(y) dy (∗) 1 is analytic. Assuming that this is true, we have by equations (64) and (65)

s 1 − s 1 1 F(s) = G + G + − . 2 2 s − 1 s which shows that F is analytic for all s ∈ C apart from simple poles at s = 1 and s = 0, and it completes the proof of theorem 3.3! So let us prove the claim (∗): Write G(z) = P G (z), where n∈N n Z n+1 z Gn(z) := y g(y) dy. n

We will prove that the Gn are all analytic functions on all of C. Consider the diﬀerence quotient for Gn(z) (exactly as in the standard proof for the analytic continuation of zeta):

1 Z n+1 1 Z n+1 yz yh − 1 g(y) dy = yzg(y)(1 + h log(y) + ρ(h, y) − 1) dy h n h n where ρ(h, y) = O(h2) for bounded values of y. So one may divide by h and take the limit h → 0.

44 Next, we want to prove that the partial sums of the Gn converge uniformly on a suitable domain. Consider z in the half-plane <(z) < K for some ﬁxed K. There Z ∞ Z ∞ z K y g(y) dy ≤ y |g(y)| dy. (66) N N Now we estimate |g(y)|:

∞ −πy X 2 X e |g(y)| = e−πn y ≤ e−πny = . 1 − e−πy n∈N n=1 The denominator is clearly bounded below for 1 < y, so the right-hand side of inequality (66) is ﬁnite for e. g. N = 1. As an immediate consequence, the integrals from N to inﬁnity must tend to zero, and all this was independent of s. Now we may apply theorem 2.7 and deduce that G is analytic on the half-plane <(s) < K. Since K was arbitrary, G is analytic on the whole complex plane. 2

45 3.4 The Gamma Function Revisited 22.2.99 We have seen that ζ and Γ go together like Laurel & Hardy, Batman & Robin, or Hardy & Wright. We need to know properties of Γ (e. g. Γ(s) 6= 0 for all s) in order to understand ζ better.

Theorem 3.11 (W like Weierstrass) Deﬁne a function f(s) by

∞ γs Y h s − s i f(s) = se 1 + e n (67) n 1 where γ is the EULER-MASCHERONI constant. Then f(s) is an analytic function on the whole of the complex plane, and it has zeros in 0, −1, −2, −3,... only.

Proof of theorem W: Consider the function

∞ ∞ X X h s s i g(s) := g (s) := log 1 + − . (68) n n n 1 1

Each gn is analytic away from −1, −2, −3,.... We want to prove that the series of the gn converges uniformly on {s ∈ C : |s| < K} for every ﬁxed K > 0. Choose N > 2K, so for all n ≥ N, |s/n| ≤ 1/2, and

s s 1 s 2 1 s 3 log 1 + = − + − ... n n 2 n 3 n

Thus we can estimate gn(s) for all these s and n: 1 s 2 1 s 3 |g (s)| ≤ + + ... n 2 n 3 n |s|2 1 |s|2 2K2 ≤ ≤ 2 < . n2 |s| n2 n2 1 − n We can sum this for n = N to inﬁnity:

∞ ∞ X X 2K2 g (s) ≤ , n n2 n=N N

46 and the latter is arbitrarily small if N is large, as the tail end of a convergent series. So the series of the analytic functions gn(s) converges uniformly on |s| < K for arbitrary K. By theorem 2.7, we deduce that the limit g(s) is analytic for all s not equal to −1, −2, −3,.... The same holds for ∞ Y h s − s i exp(g(s)) = 1 + e n . n 1 After multiplying this by seγs, we see that f(s) is an analytic function away from −1, −2, −3,... It is clear that f has zeros in all these points, so that log(f(s)) has a singularity there. Conversely, for all s away from these obvious zeros, we have shown that log(f(s)) is analytic, so that f cannot be zero elsewhere. Finally, for some fixed m ∈ N, consider the infinite product defining f without the factor corresponding to n = m. The same estimates as above show that the log of this is analytic in s = −m, so f is analytic in s = −m as well. 2

Corollary 3.12 The zeros of f as in theorem 3.11 are all simple, and the function ∞ −1 1 1 −γs Y s s = e 1 + e n f(s) s n 1 is analytic on C apart from simple poles at 0, −1, −2,.... The function 1/f has no zeros at all (because f has no poles). Theorem 3.13 (E like Euler) For all s 6= 0, −1, −2,... holds ∞ s 1 1 Y 1 s −1 = 1 + 1 + . f(s) s n n 1 Proof: We insert the deﬁnition of the Euler constant γ. N s(1+ 1 +...+ 1 −log(m)) Y s − s f(s) = s lim e 2 m lim 1 + e n m→∞ N→∞ n 1 m s(1+ 1 +...+ 1 −log(m)) Y s − s = s lim e 2 m 1 + e n m→∞ n 1 m Y s = s lim m−s 1 + . (69) m→∞ n n=1

47 Now comes a little rabbit out of a hat: Write m in a complicated way!

1 1 1 m = 1 + 1 + · ... · 1 + . (70) 1 2 m − 1

Insert this into equation (69) and use the fact that for all s

1 s lim 1 + = 1. (71) m→∞ m

Thus equation (69) becomes

m−1 −s m Y 1 Y s f(s) = s lim 1 + 1 + m→∞ n n n=1 n=1 s m −s 1 Y 1 s = s lim 1 + 1 + 1 + m m n n 1 m −s Y 1 s = s lim 1 + 1 + . (72) m n n 1 Now invert both sides, and the proof of theorem E is complete. 2

Corollary 3.14 For all s ∈ C holds 1 1 · 2 · ... · (m − 1)ms = lim . f(s) m→∞ s(s + 1) · ... · (s + m − 1)

Proof: By theorem E, for s 6= 0, −1, −2,...

m−1 s 1 1 Y 1 s −1 = lim 1 + 1 + f(s) m s n n 1 s 1 s 1 s 1 (1 + 1) 1 + ... 1 + = lim 2 m−1 m s s s 1 + 1 ... 1 + m−1 1 s 1 s 1 (1 + 1)2( 1 + ... (m − 1) 1 + = lim 2 m−1 m s (1 + s)(2 + s) ... (m − 1 + s)

48 where we have just expanded the fraction by 2 · 3 · ... · (m − 1). Now collect the integers in the numerator into one product, and the other factors into a product m−1 s Y 1 1 + = ms n n=1 by the trick we did in equation (70). This completes the proof of the corollary. 2

49 Theorem 3.15 For all s such that <(s) > 0, 23.2.99

1 Z ∞ = Γ(s) = e−tts−1 dt. (73) f(s) 0 Thus, we have three representations of the Gamma function - the deﬁnition, and the ones given in theorems W and E. And they are all useful!

Proof of theorem 3.15: Deﬁne for integers n ∈ N Z n n t s−1 Γn(s) := 1 − t dt. 0 n

Evaluate Γn(s) by integrating by parts and a substitution t = uτ, which means dt = u dτ. Z 1 s n s−1 Γn(s) = n (1 − τ) τ dτ 0 τ s 1 nsn Z 1 = ns (1 − τ)n + (1 − τ)n−1τ s dτ s 0 s 0 nsn(n − 1) Z 1 = (1 − τ)n−2τ s+1 dτ = ... s(s + 1) 0 nsn · (n − 1) · (n − 2) · ... · 2 · 1 = . s(s + 1) · ... · (s + n)

Now let n tend to inﬁnity, and use the corollary 3.14 which shows 1 lim Γn(s) = . (74) n f(s)

To complete the proof of theorem 3.15, we want to prove

lim Γn(s) = Γ(s). (75) n This is plausible, because

t n lim 1 − = e−t (76) n n

50 for all t (to prove this, just take logs, replace 1/n by h, and apply l’Hopital’s rule). BUT to apply equation (76) to our problem, an exchange of limit and integral is required! We must prove

Z n t n lim e−t − 1 − ts−1 dt = 0. (77) n 0 n So we estimate the integrand in equation (77): n n s−1 −t t σ−1 −t t t e − 1 − = t e − 1 − . (78) n n We need the following estimate:

n 2 −t −t t t e e − 1 − ≤ (79) n n for all t ∈ [0, n]. Prove this as an exercise, or see Whittaker & Watson, A Course in Modern Analysis - one of the all-time classics! Assuming this estimate, we get

Z n n Z ∞ σ−1 −t t 1 −t σ+1 Γ(σ + 2) t e − 1 − dt ≤ e t dt = , (80) 0 n n 0 n which obviously tends to zero (note that the convergence is even uniform for bounded s, although we do not need this here). 2

Corollary 3.16 For all s ∈ C, s not in N, we have π Γ(s)Γ(1 − s) = . sin(πs)

Proof: By theorem W,

∞ ∞ −1 −1 −1 Y s − s Y s s Γ(s)Γ(−s) = 1 + e n 1 − e n s2 n n 1 1 ∞ −1 −1 Y s2 −π = 1 − = s2 n2 s sin(πs) 1

51 using the formula ∞ Y s2 sin(πs) = πs 1 − . (81) n2 1 Now the corollary follows, because −sΓ(−s) = Γ(1 − s). 2

Instead of proving formula (81), we do an application: Look at (πs)3 sin(πs) = πs − + ... (82) 6 By equation (81), this is equal to 1 1 1 πs 1 − s2 + + + ... + ... 1 4 9 ∞ X 1 = πs − πs3 + ... n2 1 Comparing the coeﬃcient at s3 with that of equation (82), we have proved

∞ X 1 π2 = . n2 6 n=1 See also exercise 20, proving that ζ(2k) is a rational multiple of π2k. In contrast to this, we know very little about the values ζ(3), ζ(5),... Apery proved only in 1978 that ζ(3) 6∈ Q. See the article ‘A Proof that Euler Missed’ by A. van der Poorten (Mathematical Intelligencer 1979) or ‘Roger Ap´eryet l’irrationnel’ by Michel Mendes-France. For information on the computational evidence for the Riemann Hypothesis, see A. Odlyzko’s homepage http:// www.research.att.com/ awo or the article ‘Primes, Quantum Chaos and Computers’ by A. Odlyzko (Num- ber Theory, NRC(1993), 35-46.

Theorem 3.17 Deﬁne N(T ) to be the number of zeros of the Riemann zeta function up to height T ,

N(T ) := #{s ∈ C : 0 ≤ <(s) ≤ 1, ζ(s) = 0, 0 < =(s) < T }.

52 We do not know a formula for each individual zero, but we know an asymptotic formula: T T T N(T ) = log − + O(log(T )). 2π 2π 2π

We will prove this later, perhaps. Note however, that the proof makes use of Stirling’s general formula

1 log(Γ(s)) = −s + s − log(s) + O (1), (83) 2 δ provided |Arg(s)| < π − δ.

53 Figure 3: Primes modulo 4

p ≡ 1 (mod 4) 5 13 17 29 37 41 p ≡ 3 (mod 4) 3 7 11 19 23 31

4 Primes in an Arithmetic Progression 26.2.99 We start oﬀ with two elementary results, then give more sophisticated proofs of them, suggesting a general method. The algebraic requirements to this method are characters of abelian groups, the analytic part is a non-vanishing statement about L-functions. The culminating point is Dirichlet’s general Theorem 4.3 in section 4.2.

4.1 Two Elementary Propositions Consider all the primes congruent to 1 modulo four (see figure 3, first row) resp. congruent to three modulo four (second row). We might guess that there are infinitely many primes of each type.

Proposition 4.1 There are inﬁnitely many primes congruent to three modulo four.

Proof 1: This proceeds like the first proof for the infinity of prime numbers. Suppose that the proposition is false, and there are only r such primes p1, . . . , pr. Define 2 N := (p1 · ... · pr) + 2. 2 2 Since p1 ≡ ... ≡ pr = 1 (mod 4), we have N ≡ 3 (mod 4). Now N decom- poses into prime factors, N = q1 · ... · qk, which must all be odd. So they are all congruent to 1 or 3 modulo four. At least one of the primes qi must be congruent to 3, since otherwise N would be congruent to 1. So qi is one of p1, . . . , pr and divides N and N − 2, hence divides 2, a contradiction. 2

Proposition 4.2 There are inﬁnitely many primes congruent to 1 modulo four.

54 Proof: This proof is slightly diﬀerent. Rather than deriving a contradiction, we will show that for any given N > 1, there exists a prime congruent to 1 modulo four and greater than N. Given N > 1, deﬁne

M := (N!)2 + 1. (84)

Clearly, M is odd. Let p be the smallest prime factor of M, then clearly p > N. We claim p ≡ 1 (mod 4) (which completes the proof of the proposition, since N was arbitrary). To prove the claim, transform (84) into

(N!)2 = M − 1 ≡ −1 (mod p). (85)

Since p divides M, p is odd, and we may raise equation (85) to the (p−1)/2th power: p−1 p−1 (N!) ≡ (−1) 2 . (86) Now by Fermat’s Little Theorem, ap−1 ≡ 1 (mod p) for all a 6≡ 0 (mod p). So (p − 1)/2 must be even, proving the claim. 2

Consider p ≡ 5 (mod 6): One can prove that there are inﬁnitely many such primes in a way similar to the proof of Proposition 4.1. Use reasoning by absurdity and N := 2 · 3 · ... · pr + 5. What do you take for primes p ≡ 1 (mod 6) (only for 1 and 5 modulo 6, there is a chance of inﬁnitely many primes)? Exercise.

Isn’t this nice?? NO! The results are nice, but the proofs are awkward. We would like to have a general principle for proving such results.

4.2 A New Method of Proof We will re-do the proofs of Propositions 4.1 and 4.2 along the lines of proof 2 for the inﬁnity of primes. Consider for odd n ∈ N the function n−1 1 + (−1) 2 1 for n ≡ 1 (mod 4) c (n) := = (87) 1 2 0 for n ≡ 3 (mod 4)

55 This is a gadget for picking out a particular congruence class. Later, we will compare this fact with orthogonality relations for characters of abelian groups. Using the gadget, for real σ > 1,

X 1 1 X c1(p) = pσ 2 pσ p≡1 mod 4 odd p p−1 1 X 1 1 X (−1) 2 = + . (88) 2 pσ 2 pσ odd p odd p

The rearrangement here is ok, because the series involved converge absolutely. The first summand on the right of equation (88) tends to infinity as σ → 1 (see theorem 1.5). We claim that the second summand converges for σ → 1. (∗) This implies that the left hand side of (88) tends to infinity, and we conclude that there must be infinitely many primes over which the summation runs. Also, using a similar gadget,

n−1 1 − (−1) 2 0 for n ≡ 1 (mod 4) c (n) := = , (89) 3 2 1 for n ≡ 3 (mod 4) we get the existence of inﬁnitely many primes p ≡ 3 (mod 4) in very much the same way! Nice! The biggest payoﬀ is that the argument can be made to work in complete generality. At the end of this chapter, we will have proved the following theorem:

Theorem 4.3 (Dirichlet) If a ∈ N and q ∈ N are coprime, then there are inﬁnitely many primes p such that

p ≡ a (mod q).

Note that this is the most general result we could hope for. If a and q are not coprime, every number n ≡ a (mod q) will be divisible by gcd(a, q) > 1, so there can only be ﬁnitely many such primes.

56 For the moment, let us pursue the aim of another proof of Proposition 4.2, since all the essentials of Dirichlet’s proof become apparent there already. We still have to prove the claim (∗). To do this, define two functions χ, χ0 : N → {−1, 0, 1} by 0 if n even χ(n) = n−1 (−1) 2 if n odd 0 if n even χ (n) = (90) 0 1 if n odd Then define complex functions X χ(n) L(s, χ) := (91) ns n∈N and similarly L(s, χ0). Such functions are called L-functions and are a special kind of Dirichlet series. Clearly, the series defining L(s, χ) and L(s, χ0) converge absolutely for all s with <(s) > 1. Lemma 4.4 The series L(s, χ) converges for s = 1, and 1 1 1 π L(1, χ) = 1 − + − ± ... = . 3 5 7 4 Proof: Consider the integral Z 1 dt −1 1 π 2 = [tan (t)]0 = . (92) 0 1 + t 4 Substitute into this integral the expansion ∞ 1 X 1 = (93) 1 + t2 (−t2)n n=0 which converges for all 0 ≤ t < 1. Fix any 0 < x < 1, then the series in (93) converges uniformly for 0 ≤ t ≤ x. We have for all 0 < x < 1 ∞ Z x dt X Z x f(x) = = (−t2)n dt 1 + t2 0 n=1 0 ∞ X (−1)n = x2n+1, (94) 2n + 1 n=0

57 because there we may interchange integration and summation thanks to the uniform convergence. Now, we may take the limit x → 1 thanks to Abel’s Limit theorem (a nice special feature of power series - see Appendix B, since this is rather a theorem of Calculus). For x → 1, we get L(1, χ) on the right-hand side, and the integral in (92), f(1) = π/4 on the left-hand side. 2

Lemma 4.5 The functions χ and χ0 are completely multiplicative (see deﬁnition 1.19).

Proof: Check all cases for m and n modulo 4 in χ(mn) = χ(m)χ(n) resp. the same for χ0. 2

Now recall the Euler expansion for the zeta function, equation (11). Since χ and χ0 are completely multiplicative, we get in exactly the same way (see Theorem 2.1) an Euler expansion of L(σ, χ) and L(σ, χ0):

−1 Y χ(p) L(σ, χ) = 1 − pσ odd p −1 Y 1 L(σ, χ ) = 1 − (95) 0 pσ odd p As in proof 2 for the existence of inﬁnitely many primes, take logs of the two equations (95):

X χ(p) X χ(p) log L(σ, χ) = − log 1 − = + O(1), pσ pσ odd p odd p X 1 X 1 log L(σ, χ ) = − log 1 − = + O(1). (96) 0 pσ pσ odd p odd p Now add up (see equation (88)): 1.3.99

X 1 + χ(p) log(L(σ, χ ) · L(σ, χ)) = + O(1) 0 pσ odd p X 1 = 2 + O(1). (97) pσ p≡1 mod 4

58 What is the behaviour of the left-hand side as σ tends to 1 from above? π L(σ, χ) → 6= 0 4 1 X 1 L(σ, χ ) = 1 − → ∞. 0 2σ nσ n The terms O(1) in (96) are still O(1) for σ → 1, so we conclude

X 1 → ∞ pσ p≡1 mod 4 for σ → 1. This completes the second proof of Proposition 4.2. Had we subtracted instead of added the two equations (96), we would have got a proof that X 1 pσ p≡3 mod 4 diverges for σ → 1 and hence another proof of Proposition 4.1. 2

Another case: Congruences modulo 3. Consider the functions

1 if 3 6 |n χ (n) := 0 0 if 3|n   1 if n ≡ 1 (mod 3) χ(n) := −1 if n ≡ 2 (mod 3) (98)  0 if n ≡ 0 (mod 3)

As in the previous example, the functions c1 and c2 picking out a particular congruence class can be rewritten using χ and χ0.

1 1 if n ≡ 1 (mod 3) c (n) := (χ (n) + χ(n)) = 1 2 0 0 otherwise and 1 1 if n ≡ 2 (mod 3) c (n) := (χ (n) − χ(n)) = 2 2 0 0 otherwise.

Then deﬁne functions X χ(n) L(σ, χ) := nσ n

59 and similarly L(σ, χ0). As in the previous example, χ and χ0 are completely multiplicative, hence L(σ, χ) and L(σ, χ0) have an Euler product expansion. Moreover, X 1 1 L(σ, χ ) = = 1 − ζ(σ) 0 nσ 3σ 36|n tends to inﬁnity as σ tends to 1. We have all the ingredients to repeat the second proof of Proposition 4.2 in this case but one: We do not yet know whether L(1, χ) 6= 0. (99) If we knew this, we could proceed exactly as before - the key step is

X 1 log(L(σ, χ ) · L(σ, χ)) = 2 + O(1) 0 pσ p≡1 (mod 3)

As long as we do not know the fact (99), the left-hand side might have a limit as σ tends to 1. We will prove (99) only in section (4.5), as part of a general result. But let us put on record: If we want to prove results like Proposition 4.2 along the lines of the second proof, there are two things we need to get to grips with:

1. A machinery for pulling out a particular congruence class via multiplicative functions (see sections 4.3, 4.4).

2. A non-vanishing statement about L-functions at σ = 1 (see section 4.5).

60 4.3 Characters of Finite Abelian Groups In this section, we want to deal with Problem 1 from the preceding section. Consider the example n = 5. Define the functions 1 if n 6≡ 0 (mod 5) χ (n) := 0 0 if 5|n   i if n ≡ 2 (mod 5)   −1 if n ≡ 4 (mod 5) χ(n) := −i if n ≡ 3 (mod 5) (100)   1 if n ≡ 1 (mod 5)  0 if n ≡ 0 (mod 5) Now check that 1 1 if n ≡ 1 (mod 5) (χ (n) + χ(n) + χ2(n) + χ3(n)) = c (n) = 4 0 1 0 otherwise What if you want to pull out the congruence class n ≡ 2 (mod 5)? We need a general set-up. Recall that in the ring Z/n, the units are U(Z/n) = {k mod n :(k, n) = 1} (see section 1.3). This is a group under multiplication. E. g. if n = 5, U(Z/5) = {1, 2, 3, 4} is a cyclic group, generated by e. g. 2 mod 5. The multiplication table of U(Z/5) is 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 ∼ So U(Z/5) = {1, i, −1, −i}. Definition 4.6 Suppose G is a finite abelian group. A character of G is a homomorphism ∗ χ : G → (C , ·) (the multiplicative group C∗ is C−{0} equipped with the usual multiplication ·). By convention, we will write all finite groups multiplicatively in this section - hence the identity will be written as 1G or 1. For any group, the map ∗ χ0 : G → C , χ0(g) := 1 is a character, called the trivial character.

61 Lemma 4.7 If χ is a character of the ﬁnite abelian group G, then χ(1G) = 1 and χ(g) is a root of unity, i. e. χ(g)n = 1 for some n depending on g. In particular, |χ(g)| = 1 (χ(g) lies on the unit circle).

Proof: Clearly χ(1G) = χ(1G · 1G) = χ(1G)χ(1G). Then divide by the nonzero number χ(1G). As to the second statement, we use the fact that n for every g ∈ G there exists n ∈ N such that g = 1G. This implies n n χ(g) = χ(g ) = χ(1G) = 1. 2

∼ Example: G = Cn = hgi. If it helps you, G = (Z/n, +), an additive group - but if it doesn’t, just forget it. Since gn = 1, χ(g)n = 1 and χ(g) must be a n-th root of unity for this particular n! Any of the n diﬀerent n-th roots of unity can occur as χ(g), and of course χ(g) determines all the values of χ on G, since G is generated by g. So there are n distinct characters of G. We can label the characters of G with labels 0, 1, . . . , n − 1 as follows:

2πik χk is determined by χk(g) = e n m 2πikm so χk(g ) = e n . (101)

Theorem 4.8 Let G be a ﬁnite abelian group. Then the characters of G form a group under multiplication, (χ · ψ)(g) := χ(g)ψ(g), denoted Gˆ. The identity in Gˆ is the trivial character. The group Gˆ is isomorphic to G. In particular, any ﬁnite abelian group G of order n has exactly n distinct characters.

Remark 4.9 This is a lens, and through it you see into the ‘world of duality’. Think of this one day: What would happen if you took G = Z - what is

Zˆ := {homomorphisms Z → unit circle}? What if you took G = unit circle? The answer is: Fourier Analysis!

Proof of Theorem 4.8: Use the Structure Theorem for ﬁnite abelian groups, 2.3.99

62 which says that G is isomorphic to a product of cyclic groups.

k ∼ Y G = Cnj . j=1

Choose a generator gj for each of the factors Cnj and deﬁne characters on G by 2πi (j) n χ (∗,..., ∗, gj, ∗,..., ∗) := e j , i. e. ignore all entries except the j-th, and there use the same deﬁnition as in the Example above. Then the characters χ(1), . . . , χ(k) generate a subgroup of Gˆ which is isomorphic to G: Each χ(j) generates a cyclic group of order (i) nj, and this group has trivial intersection with the span of all the other χ ’s, since all characters in the latter have value 1 at gj. Likewise, for any given character of G, it is easy to write down a product of powers of the χ(j) which coincides with χ on the generators gj, hence on all of G. 2

Corollary 4.10 Let G be a ﬁnite abelian group. For any 1 6= g ∈ G, there exists χ ∈ Gˆ such that χ(g) 6= 1.

Proof: Looking again at the proof of Theorem 4.8, we may write g = r r (∗,..., ∗, gj , ∗,..., ∗) with some entry gj 6= 1, i. e. 0 < r < nj. Then 2piir χ(j)(g) = e nj 6= 1. 2

Theorem 4.11 Let G be a ﬁnite abelian group. Then for all elements h ∈ G and all characters ψ ∈ Gˆ hold the identities

X |G| if ψ = χ ψ(g) = 0 (102) 0 if ψ 6= χ0 g∈G X |G| if h = 1 χ(h) = . (103) 0 if h 6= 1 χ∈Gˆ

These identities are known as orthogonality relations for group characters.

63 Proof of the ﬁrst assertion: The case ψ = χ0 is trivial, so assume ψ 6= χ0. There must exist an element h ∈ G such that ψ(h) 6= 1. Then X X X ψ(h) ψ(g) = ψ(gh) = ψ(g), g∈G g∈G g∈G because multiplication by h only permutes the summands. But this equation P can only be true if g∈G ψ(g) = 0. Now for the second assertion, assume h 6= 1. By the Corollary 4.10, there exists some character ψ ∈ Gˆ such that ψ(h) 6= 1. We play the same game: X X X ψ(h) χ(h) = (ψ · χ)(h) = χ(h), χ∈Gˆ χ∈Gˆ χ∈Gˆ since multiplication by ψ only permutes the elements of Gˆ, and again this P can only be true if χ∈Gˆ χ(h) = 0. 2

Corollary 4.12 For all g, h ∈ G, we have

X |G| if g = h χ(g)χ(h) = 0 if g 6= h χ∈Gˆ

Proof: Note that χ(h−1) = χ(h)−1 = χ(h), since χ(h) is on the unit circle. Then use Theorem 4.11 with gh−1 in place of h. 2

∼ This is the gadget in its ultimate version! As an example, take G = U(Z/5) = C4. Here is a table of all the characters.

χ0 χ1 χ2 χ3 1 1 1 1 1 2 1 i −1 −i (104) 4 1 −1 1 −1 3 1 −i −1 i

64 Note that we have written the elements of U(Z/5) in an unusual ordering - this is given by 20, 21, 22, 23. The character values behave likewise. Note also 2 3 −1 χ1 = χ2 and χ1 = χ3 = χ1 . We have used earlier

χ0(n) + χ1(n) + χ2(n) + χ3(n) = 4c1(n) which is just the case h = 1 of the corollary 4.12. We asked then, what about c2(n) (1 if n congruent to 2 and 0 otherwise)? The corollary says take h = 2, and we get χ0(n) − iχ1(n) − χ2(n) + iχ3(n) = 4c2(n). Check the cases!

Remark 4.13 If you remember something about Fourier Analysis - this is really like Fourier Analysis, only nicer. The rule 1 X f(g)g(h) |G| h∈G is an inner product on the vector space of all functions on G, and the characters form a complete orthonormal set. There are no worries about convergence, integrability. . . And in particular, any complex function on G can be written as linear combination of the characters.

4.4 Dirichlet characters and L-functions Deﬁnition 4.14 Given 1 < q ∈ N, consider G := U(Z/q) and a character χ ∈ Gˆ. Extend χ to a function X on N by setting χ(n mod q) if n coprime to q X(n) := (105) 0 otherwise

Then the function X is called a Dirichlet character modulo q. Note that this is a slight abuse of language - it is not meant to say that N were a group. However, we will even write χ instead of X for the Dirichlet character associated to χ. In the same way, for any a ∈ G, we can extend the function

1 if b = a c (b) := (106) a 0 otherwise

65 to a periodic function on N, which will also be written as ca. Finally, associate to each Dirichlet character χ the function

∞ X χ(n) L(s, χ) := , ns n=1 called the L-function of χ.

Example: Take the trivial character χ0 of U(Z/4). The associated Dirich- let character is just the function χ0 that we used in the second proof of Propositions 4.2 and 4.1. And the functions c1 and c3 that we used there are extensions of functions on U(Z/4) as in (106). Same thing for the L-functions - the notation has been carefully chosen to be consistent.

Proposition 4.15 A Dirichlet character is completely multiplicative, and the associated L- function has an Euler product expansion.

Proof: Let χ be a Dirichlet character modulo q. If two integers m, n are given, and at least one of them is not coprime to q, then neither is the product mn. So χ(mn) = 0 = χ(m)χ(n). If on the other hand, both m and n are coprime to q, then (m mod q) · (n mod q) = (mn mod q) by deﬁnition, and because χ in the original sense is a group character, we have χ(mn) = χ(m)χ(n). The existence of an Euler product expansion follows then directly from Theorem 2.1. Since χ(p) = 0 for all p dividing q, we get

−1 −1 Y χ(p) Y χ(p) L(s, χ) = 1 − = 1 − (107) ps ps p p6 |q 2

Clearly the L-functions converge for <(s) > 1, by comparison with the Rie- mann zeta function. Now let us see how these L-functions can be marshalled to prove Dirichlet’s Theorem about primes in an arithmetic progression. By Theorem 2.1 which gives the Euler product expansion, L(s, χ) 6= 0 for

66 <(s) > 1. So we may take logs in (107) and expand the log:

∞ X χ(p) X X 1 χ(pm) log L(s, χ) = − log 1 − = ps m psm p6 |q p6 |q m=1 X χ(p) = + O(1) (108) ps p6 |q as before. For a given congruence class a mod q, with a coprime to q, multiply both sides by χ(a) and sum over all χ ∈ U\(Z/q) resp. the associated Dirichlet characters. We get

X X X χ(p) χ(a) log L(s, χ) = χ(a) + O(1) (109) ps χ χ p6 |q

Since the series on the right converges absolutely, we may interchange summations. By corollary 4.12

X φ(q) if p = a (mod q) χ(a)χ(p) = 0 otherwise. χ

There, φ(q) = |U(Z/q)| by deﬁnition (the Euler phi function, see (19) in section 1.3). We have proved now

X X 1 χ(a) log L(s, χ) = φ(q) + O(1). (110) qs χ p≡a mod q

Now drumroll please! Let s → 1. We claim

i) The L-function L(s, χ0) has a simple pole at s = 1.

ii) For all χ 6= χ0, the L-function L(χ, s) has a nonzero limit as s → 1. Once these claims are proved, we know that the left-hand side in (110) tends to infinity as s → 1. For the right-hand side, this means that there must be infinitely many summands. This will complete the proof of Dirichlet’s Theorem. The first claim is quite easy to prove: 5.3.99

67 −1 Y 1 Y 1 L(s, χ ) = 1 − = 1 − ζ(s), (111) 0 ps ps p6 |q p|q and we know that ζ has a simple pole at s = 1. The second claim is horribly diﬃcult to prove! A last remark before we embark on this. Looking back at ﬁgure 3, one might have guessed that both congruence classes of primes modulo four contain ‘about’ the same number of primes up to a given bound. This is true in complete generality. For all a coprime to q,

#{p ≡ a (mod q), p ≤ T } 1 lim → . (112) T →∞ #{p ≤ T } φ(q)

Note that the limit is independent of a. This can be proved by a slight reﬁnement of the methods given in this chapter.

68 4.5 Analytic continuation of L-functions and Abel’s Summation Formula

In this section, we will not only complete the proof of Dirichlet’s Theorem 4.3. On the way, we will see Abel’s Summation Formula and and the analytic continuation of L-functions to the half-plane <(s) > 0.

Theorem 4.16 (Abel) P Let a(n) be an arithmetical function and deﬁne A(x) := n≤x a(n). Let f : [y, x] → C be diﬀerentiable with continuous derivative. Then X Z y a(n)f(n) = A(y)f(y) − A(x)f(x) − A(t)f0(t) dt. (113) x

Proof: Assume x, y ∈ N are integral for ease, m = y and k = x. So the left-hand side of (113) is

m m X X a(n)f(n) = (A(n) − A(n − 1))f(n) n=k+1 n=k+1 m X = − A(n − 1)(f(n) − f(n − 1)) + A(m)f(m) − A(k)f(k). n=k+1 This is rather like integration by parts - sums instead of integrals, taking differences instead of differentiating. And there are some terms coming from the boundary of the summation interval, too! Note that we did not use the hypothesis that f be differentiable up to now. But we are not finished yet. Using A(t) = A(n) for all t ∈ [n, n + 1[, we get Z n Z n A(n − 1)(f(n) − f(n − 1)) = A(n − 1) f0(t) dt = A(t)f0(t) dt. n−1 n−1 Sum this from n = k + 1 to n = m to get the statement of the theorem. 2

Apply this formula to a(n) := χ(n)(χ a Dirichlet character modulo q, but −s Pk+q−1 not the trivial character) and f(t) := t . We have n=k χ(n) = 0 for all k ∈ N. Hence A(x) = O(1), in fact |A(x)| ≤ φ(q) for all x. Abel’s summation formula gives X χ(n) A(y) Z y A(t) = − 1 + s dt. (114) ns ys ts+1 1

69 The integral on the right-hand side of equation (114) can be split into integrals from 1 to 2, from 2 to 3 and so on. The series of these integrals converges uniformly for <(s) > δ, with any fixed δ > 0, hence we may let y → ∞. And each of these integrals is an analytic function of s for <(s) > 0. Hence the function L(s, χ) is analytic in this domain by Theorem 2.7. This argument is in fact the same that we used in one of the proofs of the analytic continuation of the zeta function. We will now finally prove L(s, ψ) 6= 0 for s = 1. Consider first the case that ψ is a non-real Dirichlet character, i. e. not all ψ(n) are ±1. Consider for σ > 1 Y X X X χ(p) log L(σ, χ) = log L(σ, χ) = − log 1 − (115) ps χ χ χ p6 |q X X X χ(p)m = . mpσm χ p6 |q m Suppose L(1, ψ) = 0. Then we must also have L(1, ψ¯) = 0. By hypothesis, ψ 6= ψ¯, and both ψ and ψ¯ appear in the product over all characters in (115). As σ tends to 0, the simple pole of L(s, χ0) is doubly cancelled by the zeros in L(s, ψ) and L(s, ψ¯), hence the product must tend to 0 and the logarithm in (115) to −∞. But the right-hand side of (115) is always nonnegative! P m This follows from χ χ(p ) = 0 or φ(q) (see Theorem 4.11). This is a contradiction, and we have proved L(1, ψ) 6= 0 in the case that ψ is not real. The case that ψ is real, i. e. ψ(n) = ±1 for all n ∈ N, is rather more 8.3.99 complicated. Suppose again L(1, ψ) = 0. Then ζ(s)L(s, ψ) must be analytic in the half-plane <(s) > 0. Write F(s) := ζ(s)L(s, ψ) as Dirichlet series (see Application 1.27), ∞ X f(n) F(s) := ns n=1 where the function f := ψ ∗ u is defined by X f(n) = (ψ ∗ u)(n) = ψ(d). d|n Lemma 4.17 Define another arithmetical function g by 1 if n is a square g(n) := 0 otherwise.

70 Then f(n) ≥ g(n) for all n ∈ N. Proof of the Lemma: Note that both f and g are multiplicative arithmetical functions. So it is enough to consider the case n = pk, a prime power. We have   1 if ψ(p) = 0  k + 1 if ψ(p) = 1 f(pk) = 1 + ψ(p) + ··· + ψ(p)k =  0 if ψ(p) = −1 and k odd  1 if ψ(p) = −1 and k even. Clearly then f(n) ≥ 0 for all n. This settles already the claim of the lemma in the case that n is not a square. If n is a square, the exponent of each prime in n is even, and we get f(n) ≥ 1 by looking at the table above. This completes the proof of Lemma 4.17. 2 Back to the main proof: Fix 0 < r < 3/2. Since F is analytic in half-plane σ > 0, look at Taylor expansion of F about s = 2. ∞ X F(ν)(2) F(2 − r) = (−r)ν, ν! ν=1 where the ν-th derivative F(ν)(2) is given by ∞ X (− log n)ν F(ν)(2) = f(n) . (116) n2 n=1 We will prove later that the Dirichlet series for F converges uniformly for σ > 0, so that we may indeed diﬀerentiate term by term. Now consider a general summand of the Taylor expansion. ∞ ∞ F(ν)(2) rν X f(n)(log n)ν rν X g(n)(log n)ν (−r)ν = ≥ ν! ν! n2 ν! n2 n=1 n=1 ∞ ∞ rν X 1 · (log(n2))ν (−2r)ν X (− log(n))ν = = ν! n4 ν! n4 n=1 n=1 (−2r)ν = ζ(ν)(4). ν! Use this inequality for all terms of the Taylor expansion of F. X (−2r)ν F(2 − r) ≥ ζ(ν)(4) = ζ(4 − 2r). (117) ν! ν=0

71 Then let r → 3/2. The right-hand side of equation (117) tends to inﬁnity, since ζ has a pole at s = 1. The left-hand side is bounded because F(s) is analytic for s > 0, a contradiction. We still have to prove our claim that the Dirichlet series for F does converge uniformly for all s > 0. Look again at equation (116) and plug it into the Taylor series for F about s = 2.

∞ ∞ X 1 X (log n)ν F(2 − r) = rν f(n) . ν! n2 ν=0 n=1 Note that the minus sign of −r cancels with that in the derivative, so that all terms are positive. Hence we may interchange the summations,

∞ ∞ X f(n) X 1 F(2 − r) = (r log n)ν (118) n2 ν! n=1 ν=0 and this sum converges for all r with |r| < 2, because by our assumption, F(s) is analytic in the whole half-plane <(s) > 0. The inner sum in equation (118) is just er log n = nr. So equation (118) becomes

∞ X f(n) F(2 − r) = . (119) n2−r n=1 The right-hand side converges for all r < 2, and if we substitute s = 2−r, we get just the Dirichlet series for F back again, which converges for all s0 > 0. Since any Dirichlet series convergent for all s ≥ s0 converges uniformly in that domain, the proof is complete. 2

72 A Proof of Theorem 3.7

Let us ﬁrst record a few lemmata which are of interest in their own right. However, this is rigorously the shortest rigorous proof that I know - no un- necessary digressions intended.

Lemma A.1 Consider the sequence of functions

K X 2πikx DK (x) := e , for K ∈ N, (120) k=−K called the Dirichlet kernel. It has several nice properties:

Z 1 DK (x) dx = 1 (121) 0 sin((2K + 1)πx) D (x) = (122) K sin(πx) Z 1 K X 2πiky g(y + x)DK (x) dx = cke , (123) 0 k=−K where ck are the Fourier coeﬃcients of g as in theorem 3.7.

R 1 2πikx Proof of lemma A.1: Property (121) follows from 0 e dx = 0 for all k 6= 0. Property (122) is proved by induction on k or directly by summation of a geometric progression. Property (123) follows from

Z 1 Z y g(y + x)DK (x) dx = g(z)DK (z − y) dz 0 y−1 Z 1/2 = g(z)DK (y − z) dz. −1/2

In the last step, we have used that g and DK are periodic functions, and that DK is an even function. At this point, we put in the deﬁnition of the DK , interchange integral and sum, and extract a factor e2πiky from each summand, which clearly gives the right-hand side of equation (123). 2

73 Lemma A.2 (Riemann-Lebesgue Lemma) Let g be a continuous periodic function and ck be the Fourier coeﬃcients of g. Then lim ck = 0. |k|→∞ Proof of lemma A.2: Deﬁne for all continuous complex-valued periodic functions u, v the inner product Z 1 (u, v) := u(x)v(x) dx 0 and the norm kuk := p(u, u). 2πikx Let uk(x) = e , so that ck = (g, uk). Using the linearity of (., .) and

(uk, ul) = 0 for all k 6= l, (uk, uk) = 1 for all k ∈ Z, we get K PK 2 2 X 2 kg − k=−K (g, uk)ukk = kgk − |(g, uk)| . k=−K Since the left-hand side is nonnegative, the sum on the right-hand side must be bounded independent of K - so actually the series P |c |2 converges! As Z k a consequence, we get that the ck tend to zero. 2

Corollary A.3 Since Z 1 ck + c−k = 2i g(x) sin(Kx) dx, 0 the latter integrals also tend to zero as K tends to inﬁnity. Now we are ready to complete the proof of theorem 3.7: By lemma A.1, the partial sums of the Fourier series are given by the left- hand side of (123). We manipulate this integral a little, using equation (122): Z 1 Z 1/2 g(y + x) − g(y) g(y + x)DK (x) dx = sin((2K + 1)x) dx 0 −1/2 sin(πx) Z 1/2 + g(y)DK (x) dx (124) −1/2

74 The last integral in equation (124) simply equals g(y) for all K by property (121) of the Dirichlet kernel. For the ﬁrst summand, observe that

g(y + x) − g(y) sin(πx) is a periodic continuous function for x ∈ [−1/2, 1/2] (the limit in x = 0 exists by l’Hopital’s rule). By the Riemann-Lebesgue Lemma A.2, or rather its corollary, this implies that the ﬁrst summand tends to zero as K tends to inﬁnity. 2

75 B Abel’s Limit Theorem

Theorem B.1 (Abel) Given a real power series

∞ X n f(x) := anx 0 which converges for all x with 0 < x < x0, suppose that the limit

∞ N X n X n L := anx0 := lim anx0 N→∞ 0 0 exists. Then the limit of f(x) exists as x tends to x0 from below, and

lim f(x) = L. x→x0 2

Proof: For any ε > 0, we will show that for all x suﬃciently close to x0 and for all N

N X n n an(x0 − x ) ≤ ε. (125) 0 To do this, rewrite the sum in (125):

N N n X X x a (xn − xn) = a 1 − xn. (126) n 0 n x 0 0 0 0

Deﬁne y := x/x0 as a shorthand and use the geometric series expansion

1 − yn = (1 − y)(1 + y + y2 + ... + yn−1).

This gives N N n−1 X n n X X k n an(x0 − x ) = (1 − y) an y x0 . (127) 0 n=1 k=0 We may interchange the summations, since these sums are both ﬁnite.

N N N X n n X k X n an(x0 − x ) = (1 − y) y anx0 . (128) 0 k=0 n=k+1

76 The coeﬃcients of yk in equation (128) form a Cauchy sequence, since the corresponding series converges. Hence they are bounded in absolute value by some b , k N X n anx0 ≤ bk n=k+1 which depends only on k, not on N. Moreover, we know that bk tends to 0 as k tends to inﬁnity. Hence, we may choose K such that bk < ε for all k ≥ K. Then we can estimate the sum in equation (127) by splitting it:

N K−1 ∞ X n n X k X k an(x0 − x ) ≤ (1 − y) y bk + (1 − y) y ε. (129) 0 k=0 k=K

As x tends to x0, y tends to 1. This means that the ﬁrst summand on the right-hand side of equation (129) becomes arbitrarily small. The second summand is equal to yK ε, hence tends to ε, and our estimates do not depend on N anymore. 2