ELEMENTS OF ANALYTIC NUMBER THEORY
P. S. Kolesnikov, E. P. Vdovin
Lecture course
Novosibirsk, Russia 2013 Contents
Chapter 1. Algebraic and transcendental numbers4 § 1.1. Field of algebraic numbers. Ring of algebraic integers4 1. Preliminary information4 2. Minimal polynomial6 3. Algebraic complex numbers9 4. Algebraic integers 11 § 1.2. Diophantine approximations of algebraic numbers 13 1. Diophantine approximation of degree ν 13 2. Dirichlet approximation theorem 15 3. Liouville theorem on Diophantine approximation of algebraic numbers 18 § 1.3. Transcendentality of e and π 20 1. Hermite identity 20 2. Transcendentality of e 23 3. Symmetrized n-tuples 25 4. Transcendentality of π 26 § 1.4. Problems 29
Chapter 2. Asymptotic law of distribution of prime numbers 30 § 2.1. Chebyshev functions 31 1. Definition and estimates 31 2. Equivalence of the asymptotic behavior of Chebyshev functions and of the prime-counting function 32 3. Von Mangoldt function 34 § 2.2. Riemann function: Elementary properties 35 1. Riemann function in Re z > 1 35 2. Distribution of the Dirichlet series of a multiplicative function 36 3. Convolution product and the M¨obiusinversion formula 37 4. Euler identity 38 5. Logarithmic derivative of the Riemann function 39
2 Contents 3
6. Expression of the integral Chebyshev function via the Riemann function 40 § 2.3. Riemann function: Analytic properties 43 1. Analytic extension of the Riemann function 43 2. Zeros of the Riemann function 47 3. Estimates of the logarithmic derivative 48 4. Proof of the Prime Number Theorem 51 § 2.4. Problems 55
Chapter 3. Dirichlet Theorem 56 § 3.1. Finite abelian groups and groups of characters 56 1. Finite abelian groups 56 2. Characters 58 3. Characters modulo m 60 § 3.2. Dirichlet series 60 1. Convergence of L-series 60 2. Landau Theorem 66 3. Proof of the Dirichlet Theorem 68 Chapter 4. p-adic numbers 71 § 4.1. Valuation fields 71 1. Basic properties 71 2. Valuations over rationals 74 3. The replenishment of a valuation field 76 § 4.2. Construction and properties of p-adic fields 79 1. Ring of p-adic integers and its properties 80 2. The field of p-adic rationals is the replenishment of rationals in p-adic metric 81 3. Applications 84 § 4.3. Problems 85 Bibliography 86
Glossary 87 Index 88 CHAPTER 1
Algebraic and transcendental numbers
§ 1.1. Field of algebraic numbers. Ring of algebraic integers 1. Preliminary information. Let us recall some basic notions from Abstract Algebra. Throughout we use the following notations: P is the set of all prime numbers; N is the set of positive integers (the set of natural numbers); Z is the set of all integers; Q is the set of all rational numbers; R is the set of all real numbers; C is the set of all complex numbers, C = R + ιR, ι2 = −1. Given a field F , symbol F [x] denotes the ring of polynomials in variable n x with coefficients in F . If f(x) = a0 + a1x + ··· + anx ∈ F [x], ai ∈ F , is chosen so that an 6= 0, then n is called the degree of f(x), it is denoted by deg f, while an ∈ F is called the leading coefficient of f(x), and if an = 1 then f(x) is called monic. If f(x) = 0 (all coefficients are equal to zero) then the degree of f(x) is said to be −∞. If f(x), g(x) ∈ F [x], g(x) 6= 0, then there exist unique q(x), r(x) ∈ F [x] such that f(x) = g(x)q(x) + r(x), deg r < deg g. (1.1)
These polynomials (quotient q(x) and remainder r(x)) can be found by the well-known division algorithm. If r(x) = 0 then we write g | f (g divides f). One may easily note the similarity between division algorithms in the ring of integers Z and in the ring of polynomials F [x]. Indeed, these are par- ticular examples of Euclidean rings, and there are many common features and problems that can be solved in similar ways for integers and polynomi- als. In particular, the greatest common divisor (gcd) d of two polynomials f, g ∈ F [x] is defined as a monic common divisor which is divided by every
4 § 1.1. Field of algebraic numbers 5 other common divisor, i.e., d = gcd(f, g) if and only if d | f, d | g, and for every h ∈ F [x] with h | f and h | g it follows that h divides d. To find gcd of f and g, one may use the Euclidean algorithm based on the following observation: If f and g are related by (1.1) then gcd(f, g) = gcd(g, r). Moreover, if d = gcd(f, g) then there exist p(x), s(x) ∈ F [x] such that f(x)p(x) + g(x)s(x) = d(x).
Exercise 1.1. Let f1, . . . , fn ∈ F [x] be a finite family of polynomials over a field F . Prove that there exists a unique monic greatest common divisor of f1, . . . , fn. Suppose R is a commutative ring with an identity (e.g., R = Z or R = F [x] as above). A subset I ⊆ R is called an ideal of R if a ± b ∈ I for every a, b ∈ I, and ax ∈ I for every a ∈ I, x ∈ R. For example, the set of all even integers is an ideal of Z; the set {f(x) ∈ F [x] | f(α) = 0} is an ideal of F [x], where α is an element of some extension field of F . Since an intersection of any family of ideals is again an ideal, for every set M ⊆ R there exists minimal ideal of R which contains M, it is denoted by (M). It is easy to note that X (M) = xiai | xi ∈ R, ai ∈ M . i An ideal I of R is said to be principal if there exists a ∈ R such that I = (a), where (a) stands for ({a}). Recall that a commutative ring R is called an integral domain (or simply a domain) if ab = 0 implies a = 0 or b = 0 for all a, b ∈ R. In particular, Z and F [x] are integral domains. An integral domain R such that every ideal of R is principal is called a principal ideal domain. Exercise 1.2. Prove that Z and F [x] (where F is a field) are principal ideal domains. In particular, if f(x), g(x) ∈ F [x] then ({f, g}) = (gcd(f, g)). If R is a domain, then we can consider the field of fractions of R. In order to construct it we start with the Cartesian product R × (R \{0}) of R a (here each pair (a, b) corresponds to fraction b ). Now define an equivalence relation (a1, b1) ∼ (a2, b2) ⇐⇒ a1b2 = a2b1. Let Q be the set of equivalence classes of R × (R \{0}) under this equivalence. Define the addition and multiplication on representatives by
(a1, b1) + (a2, b2) = (a1b2 + a2b1, b1b2), (a1, b1) · (a2, b2) = (a1a2, b1b2). § 1.1. Field of algebraic numbers 6
We leave for the reader to prove that all operations defined are correct and that Q is a field under these operations. Let I be an ideal of R. Then R is split into a disjoint union of congruence classes a+I = {a+x | x ∈ I}, a ∈ R, and the set of all these classes (denoted by R/I) is a ring with respect to natural operations (a + I) + (b + I) = (a + b) + I, (a + I)(b + I) = ab + I. The ring R/I obtained is called a factor ring of R over I. For example, Z/(n) = Zn, the ring of remainders modulo n. A proper ideal I of R is maximal if there are no proper ideals J of R such that I ⊂ J. For example, if R = Z then (n) is maximal if and only if n = ±p, where p is a prime natural number; if R = F [x] then (f) is maximal if and only if the polynomial f is irreducible over F . Note that if I is a maximal ideal of a commutative ring R then R/I is a field. Indeed, if a + I 6= 0 (i.e., a∈ / I) then J = {xa + b | x ∈ R, b ∈ I} is an ideal of R such that I ⊂ J. Therefore, J = R, and thus all equations of the form (a + I)X = c + I, c ∈ R, have solutions in R/I. Exercise 1.3. Prove that R[x]/(x2 + x + 1) is a field isomorphic to the field C of complex numbers. 2. Minimal polynomial. A complex number α ∈ C is algebraic if there exists a nonzero polynomial f(x) ∈ Q[x] such that f(α) = 0. A non-algebraic complex number is said to be transcendental. An algebraic number α is called an algebraic integer if there exists a monic polynomial f(x) ∈ Z[x] such that f(α) = 0. Every rational number α ∈ Q ⊂ C is obviously an algebraic one. More- over, as we will see later, a rational number is an algebraic integer if and only if it is an integer. √ √ 1 3 Exercise 1.4. (1) Prove that 2 and + ι are algebraic in- 2 2 tegers. (2) Prove that if α ∈ C is an algebraic number then Re α and Im α are algebraic numbers. Whether the same statements are true for an algebraic integer α? (3) Show that the cardinality of the set of all algebraic numbers is countable. Since the cardinality of the entire set of complex numbers is uncountable (continuum), transcendental numbers do exist. However, it is not so easy to show an example of such a number accompanied with reasonable proof. § 1.1. Field of algebraic numbers 7
Let α be an algebraic number. Denote by x a formal variable, and let
I(α, x) = {f(x) ∈ Q[x] | f(α) = 0}. It is clear that I(α, x) is an ideal of the ring Q[x]. Since Q[x] is a principal ideal domain, the ideal I(α, x) is generated by a single polynomial. Namely, the monic polynomial of minimal positive degree from h(x) ∈ I(α, x) is a generator of the ideal I(α, x), it is called the minimal polynomial for α. Given an algebraic number α, denote its minimal polynomial by hα(x) and say deg hα to be the degree of α. Lemma 1.5. Let α be an algebraic number, and let h(x) be a monic polynomial from Q[x]. Then the following conditions are equivalent:
(1) h(x) = hα(x); (2) h(α) = 0, and h divides every f ∈ I(α, x); (3) h(α) = 0, and h(x) is irreducible over Q. Proof. (1) ⇒ (2) It is obvious by definition. (2) ⇒ (3) Assume h(x) is reducible over Q, i.e., it can be decomposed into nonscalar factors as follows:
h(x) = h1(x)h2(x), where hi(x) ∈ Q[x], deg hi > 1.
Then h(α) = h1(α)h2(α) = 0, so for either of i = 1, 2 we have hi(α) = 0. Therefore, the corresponding polynomial hi(x) belongs to I(α, x), hence, hi(x) is a multiple of h(x), which is impossible due to deg hi < deg h. (3) ⇒ (1) Let hα(x) be the minimal polynomial for α. Then h(x) ∈ I(α, x) = (hα(x)), so hα | h. Since h(x) is irreducible and deg hα > 0, we have h(x) = hα(x). √ 2 For example, if √α ∈ Q then hα = x − α. For α = 2, hα = x − 2. The 1 3 number α = + ι satisfies the equation α3 + 1 = 0, but its minimal 2 2 2 polynomial is hα(x) = x − x + 1. Exercise 1.6. Prove that a minimal polynomial does not have multiple roots.
If α is an algebraic number, and β ∈ C is a root of hα(x) then β is said to be conjugate to α. Therefore, every algebraic number α of degree n has exactly n pairwise different conjugate complex numbers α1, . . . , αn, § 1.1. Field of algebraic numbers 8 including α itself. Moreover, n Y hα(x) = (x − αi) ∈ Q[x]. i=1
Given an algebraic number α, the minimal polynomial hα ∈ Q[x] is uniquely defined. If α is an algebraic integer then there also exists a monic polynomial f ∈ Z[x] of minimal degree such that hα | f. In order to prove that these f and hα coincide, we need the following observation. Let h(x) ∈ Q[x] be a monic polynomial with rational coefficients, p0 p1 pn−1 n−1 n h(x) = + x + ··· + x + x , gcd(pi, qi) = 1. q0 q1 qn−1 Denote by q(h) the least common multiple of the coefficients’ denominators:
q(h) = lcm (q0, . . . , qn−1). (1.2) Then for q = q(h) we have n−1 n qh(x) = a0 + a1x + ··· + an−1x + anx ∈ Z[x], where the gcd of all coefficients is equal to the identity: (a0, . . . , an) = 1. Indeed, assume a0, . . . , an have a common divisor d > 1. Then d | q = an, and for b = q/d ∈ Z we have bh(x) ∈ Z[x]. Therefore, qi | bpi for all i = 0, . . . , n − 1, so qi | b, and, finally, q | b = q/d, which is impossible for d > 1. Polynomials with relatively prime integral coefficients are studied in Abstract Algebra, they are called primitive. The following statement is well-known. Exercise 1.7 (Hauss Lemma ). Prove that the product of primitive polynomials is also a primitive polynomial. Proposition 1.8. Let α be an algebraic integer. Then the minimal polynomial hα(x) has integral coefficients. Proof. Suppose f(x) ∈ Z[x] be a monic polynomial with integral co- efficients such that f(α) = 0. Then Lemma 1.5 implies hα | f, i.e.,
f(x) = hα(x)g(x), g(x) ∈ Q[x].
Since both f and hα are monic polynomials, so is g. 0 Denote q = q(hα), q = q(g), where the function q(·) is given by (1.2). 0 According to the remark above, qhα(x) and q g(x) are primitive polynomials in Z[x]. However, 0 0 qq f(x) = (qhα(x))(q g(x)). § 1.1. Field of algebraic numbers 9
The Hauss Lemma implies qq0f(x) to be primitive, while f(x) has integral 0 coefficients itself. Therefore, qq = 1, i.e., all coefficients of hα and g are integral. Thus, there is no need to define separately integral minimal polynomial and degree for algebraic integers. Let us also note that for every algebraic number α there exists c ∈ N such that cα is an algebraic integer: It is deg hα enough to consider c = q(hα) .
3. Algebraic complex numbers. Let us denote the set of all alge- braic numbers by A. Recall that the Fundamental Theorem of Algebra states C to be an algebraically closed field, i.e., every non-constant polyno- mial over C has a root in C. In this section, we will prove that A ⊂ C is the minimal algebraically closed subfield of C, i.e., A is the algebraic closure of Q. Lemma 1.9. The following statements are equivalent for α ∈ C: (1) α ∈ A; (2) Q[α] := {f(α) | f(x) ∈ Q[x]} is a finite-dimensional vector space over Q; (3) Q[α] is a subfield of C.
Proof. (1) ⇒ (2). Note that f(x) = q(x)hα(x) + r(x) by the divi- sion algorithm, deg r < deg hα. Hence, f(α) = r(α), and the latter is a n−1 linear combination over Q of 1, α, . . . , α , where n = deg hα. Therefore, n−1 dim Q[α] 6 n. Moreover, 1, α, . . . , α are linearly independent since n is the minimal possible degree of a polynomial over Q annihilating α, so
dim Q[α] = deg hα. (2) ⇒ (1). It is enough to note that 1, α, α2,... are linearly dependent, so there exist a0, a1, . . . , an ∈ Q such that n a0 · 1 + a1α + ··· + anα = 0, n end at least one of ai is nonzero. Hence, h(x) = a0 +a1x+···+anx ∈ Q[x] is a nonzero polynomial annihilating α. (1) ⇒ (3) Obviously, Q[α] is a subring of C. Let 0 6= f(α) ∈ Q[α], then f ∈ Q[x] \ I(α, x). Therefore, hα does not divide f. Since hα is irreducible, we have gcd(f, hα) = 1. Hence, there exist polynomials u(x), v(x) ∈ Q[x] such that
u(x)f(x) + v(x)hα(x) = 1. § 1.1. Field of algebraic numbers 10
Then for x = α we obtain u(α)f(α) = 1, i.e., f(α)−1 = u(α) ∈ Q[α], i.e., Q[α] is a field. (3) ⇒ (1) Suppose Q[α] is a subfield of C. It is enough to consider the case when α 6= 0. Since α−1 ∈ Q[α], there exists f(x) ∈ Q[x] such that −1 α = f(α), i.e., h(α) = 0 for h(x) = xf(x) − 1, deg h > 1.
Corollary 1.10. If α1, . . . , αn ∈ A then Q[α1, . . . , αn] := {f(α1, . . . , αn) | f(x1, . . . , xn) ∈ Q[x1, . . . , xn]} is a finite-dimensional vector space over Q. Proof. For n = 1, it follows from Lemma 1.9. By induction, since dim Q[α1, . . . , αn−1] < ∞ and dim Q[αn] < ∞,
dim Q[α1, . . . , αn] 6 dim Q[α1, . . . , αn−1] · dim Q[αn] < ∞ (all dimensions are over Q).
Exercise 1.11. Prove that Q[α1, . . . , αn] is a subfield of C provided that α1, . . . , αn ∈ A. Theorem 1.12. The set of all algebraic numbers is a subfield of C. Proof. Since 1, 0 ∈ C are obviously algebraic, it is enough to prove the following two statements: (1) If α and β are algebraic numbers then α ± β and αβ are also algebraic numbers; (2) If β 6= 0 is an algebraic number then 1/β is also an algebraic number. (1) Since Q[α ± β], Q[α · β] ⊆ Q[α, β] ⊆ C, we have dim Q[α ± β] < ∞, dim Q[α · β] < ∞ by Corollary 1.10. Thus by Lemma 1.9 α ± β, αβ ∈ A. (2) By Lemma 1.9, β−1 ∈ Q[β], so Q[β−1] ⊆ Q[β] which is finite- −1 −1 dimensional over Q. Hence, dim Q[β ] < ∞ and thus β ∈ A. Theorem 1.13. The field A is algebraically closed.
Proof. Suppose α0, . . . , αn ∈ A, n > 1, αn 6= 0, ϕ(x) = α0 + α1x + n ··· + αnx ∈ A[x]. It is enough to show that ϕ(x) has a root in A. Without loss of generality, assume αn = 1. Indeed, by Theorem 1.12 we may divide ϕ(x) by αn, and the result is still in A[x]. By the Fundamental Theorem of Algebra, ϕ(x) has a root β ∈ C. Note that Q[β] ⊆ Q[α0, . . . , αn−1, β]. § 1.1. Field of algebraic numbers 11
Since βn can be expressed as a linear combination of βk, k = 0, . . . , n − 1, with coefficients depending on αi, i = 0, . . . , n − 1, as n n−1 β = −α0 − α1β − · · · − αn−1β , we have
dim Q[α0, . . . , αn−1, β] 6 n dim Q[α0, . . . , αn−1] < ∞. Hence, Q[β] is a finite-dimensional vector space over Q, and by Lemma 1.9 β ∈ A Theorems 1.12 and 1.13 imply that A is the algebraic closure of Q, which is often denoted by Q.
4. Algebraic integers. Suppose K[x1, . . . , xn] is the ring of polyno- mials in several variables over a ring K. For f ∈ K[x1, . . . , xn] denote by deg f the maximal sum of the degrees of the variables that appear in a term of f with a nonzero coefficient. Namely, f may be uniquely written as m f = f0 + f1xn + ··· + fmxn , where fi ∈ K[x1, . . . , xn−1]. Assuming deg fi are defined by induction, set deg f = max (i + deg fi). i=0,...,m Theorem 1.14. The set of all algebraic integers is a subring of the field C. Proof. Ii is enough to show that if α, β are algebraic integers then α ± β and αβ are algebraic integers as well. We will prove a more general fact: Every number of the form X k l γ = cklα β , ckl ∈ Z, k,l is an algebraic integer. Let n−1 n hα(x) = a0 + a1x + ··· + an−1x + x , m−1 m hβ(x) = b0 + b1x + ··· + bm−1x + x . By Proposition 1.8, ai, bj ∈ Z. Lemma 1.5 implies that hα and hβ are irreducible polynomials over Q, thus they have no multiple roots. Denote by α1, . . . , αn all complex roots of hα(x), and let β1, . . . , βm stand for all complex roots of hβ(x). To be more precise, set α1 = α, β1 = β. Consider the polynomial n m Y Y X k l p(x) = x − cklαi βj ∈ C[x]. i=1 j=1 k,l § 1.1. Field of algebraic numbers 12
It is clear that p(γ) = 0 and the leading coefficient of p(x) is equal to the identity. It remains to show that p(x) ∈ Z[x]. It follows from the definition of p(x) that
p(x) = f(x, α1, . . . , αn, β1, . . . , βm), where f ∈ Z[x, y1, . . . , yn, z1, . . . , zm]. Namely,
n m Y Y X k l f(x) = x − cklyi zj ∈ Z[x, y1, . . . , yn, z1, . . . , zm]. i=1 j=1 k,l
Moreover, every permutation of the variables y1, . . . , yn or z1, . . . , zm does not change the polynomial f, i.e., it is symmetric with respect to yi and with respect to zi. Hence, nm X a f(x, y1, . . . , yn, z1, . . . , zm) = ga(y1, . . . , yn, z1, . . . , zm)x , a=0 where every polynomial ga is symmetric with respect to yi and with respect to zi. On the other hand, for every a we have
X d1 dm ga(y1, . . . , yn, z1, . . . , zm) = ga,d1,...,dm (y1, . . . , yn)z1 . . . zm , d1,...,dm>1 where every polynomial ga,d1,...,dm (y1, . . . , yn) is symmetric (with respect to yi) and has integral coefficients.
Lemma 1.15. Let Ψ(y1, . . . , yn) ∈ Z[y1, . . . , yn] be a symmetric poly- nomial on y1, . . . , yn, deg Ψ = N, and let α1, . . . , αn ∈ C be the roots n of a polynomial h(x) = a0 + a1x + ··· + anx ∈ Z[x], an 6= 0. Then N an Ψ(α1, . . . , αn) ∈ Z. Proof. By the Fundamental Theorem of Symmetric Polynomials, there exists a polynomial G(t1, . . . , tn) ∈ Z[t1, . . . , tn] such that
Ψ(y1, . . . , yn) = G(σ1, . . . , σn), deg G 6 N. where σi(y1, . . . , yn) are the elementary symmetric polynomials on y1, . . . , yn. k an−k The Viet formulae imply σk(α1, . . . , αn) = (−1) , k = 1, . . . , n. Since an deg G 6 N, we obtain N N an−1 n a0 an Ψ(α1, . . . , αn) = an G − ,..., (−1) . an an The latter is an integer number. § 1.2. Diophantine approximations 13
Now we can use Lemma 1.15 for polynomials ga,d1,...,dm (y1, . . . , yn) to obtain ga,d1,...,dm (α1, . . . , αn) ∈ Z (in this case, an = 1 since αi are algebraic integers). Hence,
ga(α1, . . . , αn, z1, . . . , zm) ∈ Z[z1, . . . , zm] are symmetric polynomials on z1, . . . , zm, and by Lemma 1.15 we have
ga(α1, . . . , αn, β1, . . . , βm) ∈ Z. Therefore, p(x) ∈ Z[x].
§ 1.2. Diophantine approximations of algebraic numbers It is well-known from the course of Analysis that the set of rational numbers is a dense subset of the set of real numbers , i. e., for every Q p R α ∈ and for every ε > 0 there exists ∈ such that R q Q
p α − < ε. (1.3) q
Given a natural number N, the set QN = {p/q | p ∈ Z, 0 < q 6 N}, has 1 the following obvious property: |a − b| for all a, b ∈ . Hence, for > N 2 QN every α∈ / QN (in particular, for an irrational one), we have min |a − α| > 0. a∈QN p Therefore, in order to satisfy (1.3) for small ε, the number should have a q an unboundedly large denominator q. This observation raises the following natural question: How to measure the accuracy of a rational approximation relative to the growing denominator values?
1. Diophantine approximation of degree ν. To estimate the ac- curacy of an approximation by rationals, we will compare the difference |α − p/q| with a decreasing function q−ν , ν > 0. Namely, let us consider the following quantity: ν p εα,ν (q) = min q α − , ν > 0. (1.4) p∈Z,p/q6=α q
It turns out that the study of the behavior of εα,ν (q) as q approaches infinity leads to a necessary condition for α to be algebraic. § 1.2. Diophantine approximations 14
Definition 1.1. A real number α ∈ R possesses a Diophantine approx- imation of degree ν > 0 if
lim εα,ν (q) < ∞. (1.5) q→∞ Let us state a useful criterion that allows to determine whether a given number possesses a Diophantine approximation of a given degree. Lemma 1.16. A real number α possesses a Diophantine approximation of degree ν > 0 if and only if there exists a constant c > 0 such that the inequality
p c α − < (1.6) q qν p holds for infinitely many rationals ∈ . q Q Proof. Denote by M = M(α, c, ν) the set of all pairs (p, q) ∈ Z × N such that (1.6) holds. Suppose α possesses a Diophantine approximation of degree ν > 0. By Definition 1.1, there exists c > 0 such that
εα,ν (q) < c for infinitely many q ∈ N. Let us fix this c and note that for every such q ∈ N there exists p ∈ Z satisfying the inequality (1.6). Therefore, the set M is infinite, and there is no upper bound for the set {q | (p, q) ∈ M for some p}. It remains to show that the set {p/q | (p, q) ∈ M} is also infinite. Indeed, if it were finite then the left-hand side of (1.6) has a positive lower bound, but the right-hand side of (1.6) approaches zero since ν > 0 and q may be chosen to be as large as we need. Conversely, suppose there exists c such that (1.6) holds for infinitely many rationals p/q. Then the set M defined in the first part of the proof is infinite. Assume the set {q | (p, q) ∈ M for some p} has an upper bound N, i.e.,
{p/q | (p, q) ∈ M} ⊆ QN , where QN = {p/q ∈ Q | p ∈ Z, 0 < q 6 N}. As we have already mentioned above, for every distinct a1, a2 ∈ A the 2 inequality |a1 − a2| > 1/N holds. Hence, any infinite subset S of QN has infinite diameter, i.e., for any d > 0 one may find p1/q1 and p2/q2 in S such that |p1/q1 − p2/q2| > d. § 1.2. Diophantine approximations 15
In particular, the set S = {p/q | (p, q) ∈ M} ⊆ QN contains p1/q1 and p2/q2 such that |p1/q1 − p2/q2| > 2c. Since (1.6) holds for pi/qi, i = 1, 2, we have
p1 c p2 c α − < ν , α − < ν q1 q1 q2 q2 which implies p1 p2 1 1 2c < − < c ν + ν < 2c, q1 q2 q1 q2 a contradiction. Therefore, the set of denominators {q | (p, q) ∈ M for some p} is in- finite, so there exist infinitely many q such that εα,ν (q) < c. Hence, the sequence {εα,ν (q)}q∈N has a finite accumulation point, and (1.5) holds. Exercise 1.17. Whether a rational number possesses a Diophantine approximation of degree 1? 2. Dirichlet approximation theorem.
Theorem 1.18 (Dirichlet Approximation Theorem). For every α ∈ R and for every N ∈ N there exist p ∈ Z and q ∈ N such that
p 1 α − < , q N. q qN 6 Proof. Consider the fractional parts of the numbers kα, k = 0,...,N:
ξk = {kα} = kα − [kα] ∈ [0, 1). Divide the interval [0, 1) into N intervals of length 1/N as follows: [k/N, (k + 1)/N), k = 0,...,N − 1. According to the combinatorial Dirichlet’s Principle, when N + 1 numbers ξ0, . . . , ξN are set into N intervals, there exists at least one interval which contains at least two of these numbers, i.e.,
|ξk1 − ξk2 | < 1/N for some k1, k2, 0 6 k1 < k2 6 N. Let p = [k2α] − [k1α], q = k2 − k1. Then
p 1 1 1 α − = |α(k2 − k1) − [k2α] + [k1α]| = |ξk2 − ξk1 | < . q q q Nq § 1.2. Diophantine approximations 16
Corollary 1.19. If α ∈ R \ Q then α possesses a Diophantine approx- imation of degree ν = 2. Proof. Theorem 1.18 implies that for every natural number N there exist pN ∈ Z and qN ∈ N, qN 6 N, such that
pN 1 α − < . (1.7) qN NqN
Let us show that the sequence {qN }N>1 is not bounded. Assume the con- verse, i.e., suppose there exists a constant M such that qN 6 M for all N. Then (1.7) implies
p 1 1 min α − < 6 → 0, p/q∈QM q NqN N N→∞ which is impossible. Therefore, (1.7) holds for infinitely many numbers qN . Since qN 6 N, we have
2 pN qN α − < 1 qN for infinitely many qN .
Proposition 1.20. Let α ∈ Q. Then for every function ϕ : N → R+ p satisfying lim qϕ(q) = 0 there exist only a finite number of rationals q→∞ q such that
p α − < ϕ(q). (1.8) q Proof. Denote by S the set of all rationals p/q, gcd(p, q) = 1, satisfy- ing (1.8). Assume S is infinite. It is easy to see that the distance between any two numbers from S does not exceed 2 max ϕ(q) which is finite. Note q>1 that the set of denominators of all fractions from S has no upper bound: Otherwise, if S ⊆ QN , where QN introduced in the proof of Lemma 1.16, then S has an infinite diameter. a p Let α = . Then for all rationals except α itself the following in- b q equality holds:
a p aq − pb 1 − = . b q bq > bq Since b is fixed, we have 1 > ϕ(q), bq § 1.2. Diophantine approximations 17 which is impossible for sufficiently large q. The contradiction obtained proves that S is finite. Corollary 1.21. A rational number α does not have a Diophantine approximation of degree ν > 1. The result obtained is in some sence paradoxical: All irrational numbers possess Diophantine approximations of degree 2, but neither of rationals has a Diophantine approximation of degree ν > 1. Therefore, the existence of a Diophantine approximation of degree ν > 2 is a criterion of irrationality. Let us apply the criterion above to the base of the natural logarithm 1 1 1 e = 1 + + + ··· + + ..., (1.9) 1! 2! n! also called the Euler’s number. Proposition 1.22. The Euler’s number is irrational. Proof. Consider the sum of the first n + 1 summands in (1.9) and reduce to the common denominator: 1 1 1 p 1 + + + ··· + = n . 1! 2! n! n! Then p 1 1 1 e − n = + + ... n! (n + 1)! n + 2 (n + 2)(n + 3) 1 1 1 2 < + + ... = . (n + 1)! 2 22 (n + 1)! Define a function ϕ : N → R as follows: ϕ(1) = 1, 2 ϕ(q) = for (n − 1)! < q n!, n 2. (n + 1)! 6 > If q ∈ ((n − 1)!, n!] then 2 2 qϕ(q) n! = , 6 (n + 1)! n + 1 so lim qϕ(q) = 0. However, q→∞
p e − < ϕ(q) q § 1.2. Diophantine approximations 18 for infinitely many p/q = pn/n!, n > 2. Hence, e is irrational by Proposi- tion 1.20. 3. Liouville theorem on Diophantine approximation of alge- braic numbers. Theorem 1.23 (Liouville Theorem). Let α be an algebraic number of degree n > 2. Then α has no Diophantine approximation of degree ν > n. Proof. First, let us find a constant M > 0 such that
p M α − > (1.10) q qn for all p ∈ Z and q ∈ N. Set h(x) to be a multiple of the minmal polynomial for α with inte- ger coefficients, e.g., h(x) = q(hα)hα(x). Suppose α1, . . . , αn ∈ C are the complex roots of this polynomial, and assume α1 = α. Then n n Y Y h(x) = an (x − αk) = an(x − α) (x − αk), k=1 k=2 where an is the leading coefficient of h(x). Denote by M the following quantity: n !−1 Y M = |an| (|α| + |αk| + 1) , k=2 and let us show that (1.10) holds. If p and q meet the inequality |α − p/q| > 1 then (1.10) is valid since M < 1 (|α| > 0, n > 2). Assume p and q satisfy the condition |α − p/q| < 1. In this case,
p < 1 + |α| q and thus n p Y p |h(p/q)| = |an| α − αk − q q k=2 n p Y p p 1 α − |an| |αk| + < α − . 6 q q q M k=2 § 1.2. Diophantine approximations 19
Lemma 1.5 implies h(x) to be irreducible over Q, hence, it has no rational roots. Therefore, n p p 1 |h(p/q)| = a0 + a1 + ··· + an , q qn > qn and (1.10) follows. Finally, apply (1.10) to show that α has no Diophantine approximation of degree ν > n. Assume the converse: Let there exist ν > n and c > 0 such that
p c α − < q qν holds for infinitely many p/q ∈ Q. Then, as it was shown in the proof of Lemma 1.16, the last inequality holds for infinitely many denominators q ∈ N. Then for sufficiently large q we have c M < , qν qn in contradiction to (1.10).
The Liouville Theorem is a powerful tool that allows constructing ex- plicit examples of transcendental numbers. Namely, we obtain the following sufficient condition of transcendentality.
Corollary 1.24. Let α be a real number. If for every N ∈ N it possesses a Diophantine approximation of degree ν > N then α is transcendental. Example 1.2. The following number is transcendental: ∞ X α = 10−n!. n=1 Proof. Given N ∈ N, consider N pN X pN = 10−n! = . q 10N! N n=1 Note that ∞ pN X −n! −(N+1)! 2 α − = 10 < 2 · 10 = N+1 . qN n=N+1 qN § 1.3. Transcendentality of e and π 20
pN Therefore, for every ν > 0 there exist infinitely many rationals , N > ν, qN satisfying
pN 2 2 α − < N+1 < ν . qN qN qN Hence, α possesses a Diophantine approximation of any degree. Theo- rem 1.23 implies α is not algebraic.
§ 1.3. Transcendentality of e and π The Liouville Theorem provides a sufficient condition for a real num- ber to be transcendental. However, this condition is not necessary. There exist different methods to prove transcendentality of a series of important constants, e.g., e and π. In this section, we are going to study one of these methods known as the Hermite Method. Given an analytic function f(z) on the complex plane, denote by x Z f(z) dz
x0 the Riemann integral of f(z) along the straight segment starting at x0 ∈ C and ending at x ∈ C (the Cauchy Integral Theorem implies that this integral does not depend on the choice of a path with the same endpoints x0 and x, we choose the straight segment for convenience). 1. Hermite identity.
Lemma 1.25 (Hermite Identity). Let α ∈ C, α 6= 0, and let f(x) ∈ C[x], deg f > 1. Then for every x ∈ C we have x Z f(t)e−αt dt = F (0) − F (x)e−αx, (1.11)
0 where f(x) f 0(x) f (n)(x) F (x) = + + ··· + . α α2 αn+1 Proof. According to the Fundamental Theorem of Calculus (the Newton— Leibniz Formula), x d Z f(t)e−αt dt = f(x)e−αx. dx 0 § 1.3. Transcendentality of e and π 21
On the other hand, d F (x)e−αx = (F 0(x) − αF (x))e−αx = −f(x)e−αx. dx Hence, the derivatives with respect to x of the both sides of (1.11) coincide. It remains to compare the values at x = 0 to obtain the desired equality. The main idea of the Hermite’s method is to apply the Hermite identity (1.11) to a polynomial of the form 1 H(h(x)) = a(n−1)pxp−1h(x)p, (1.12) (p − 1)! n h(x) ∈ Z[x], n = deg h, p ∈ N. Let us establish the properties of H(h(x)). n Lemma 1.26. Let h(x) = a0 + a1x + ··· + anx ∈ Z[x], a0, an 6= 0, n > 1, and let β1, . . . , βn ∈ C be the entire collection of roots of h(x) in which every root of multiplicity k appears k times. Then for every p ∈ N, p > 2, the polynomial f(x) = H(h(x)) defined by (1.12) has the following properties: (j) (1) f (0) = 0, 0 6 j 6 p − 2; (j) (2) f (βi) = 0, 0 6 j 6 p − 1, i = 1, . . . , n; (p−1) (n−1)p p (3) f (0) = an a0; (j) (4) f (0) ∈ pZ, j > p; n P (j) (5) f (βi) ∈ pZ, j > p. i=1 Proof. It is easy to see from the construction of f(x) that 0 is its root of multiplicity p − 1 and every βi is a root of f(x) of multiplicity at least p. As we know from the Abstract Algebra course, a root of multiplicity k of a polynomial f(x) is also a root of f 0(x), f 00(x), . . . , f (k−1)(x). This implies (1) and (2). To prove (3), let us distribute all brackets in the definition of f(x) and find the term of lowest degree in x, namely, the term is 1 a(n−1)papxp−1. (p − 1)! n 0
(n−1)p p Its (p − 1)th derivative is equal to an a0. For all other terms in f(x), their (p − 1)th derivatives contain x and thus turn into zero at x = 0. Before we proceed with the proof of the remaining statements, note the following general fact. Given a polynomial Φ(x) ∈ Z[x], its jth derivative § 1.3. Transcendentality of e and π 22
Φ(j)(x), j ∈ N, belongs to j!Z[x], i.e., all nonzero coefficients of Φ(j)(x) contain the factor j!. Indeed, for all m > j we have m (xm)(j) = j! xm−j ∈ [x], j Z and the jth derivatives of xm, m < j, turn into zero. To prove (4), consider (p − 1)! Φ(x) = f(x) = xp−1h(x)p ∈ [x]. (n−1)p Z an According to the remark stated above, Φ(j)(x) ∈ j!Z[x] and thus j! f(0) ∈ ⊆ j (p − 1)!Z Z for j > p, Finally, note that n Y h(x) = an (x − βi). i=1 Hence, n anp Y anp f(x) = n (x − β )pxp−1 = n Θ(x, β , . . . , β ), (p − 1)! i (p − 1)! 1 n i=1 where n p−1 Y p Θ = x (x − yi) ∈ Z[x, y1, . . . , yn]. i=1 The polynomial n X ∂jΘ Ψ(y , . . . , y ) = (y , y , . . . , y ) ∈ [y , . . . , y ] 1 n ∂xj i 1 n Z 1 n i=1 is symmetric with respect to y1, . . . , yn, deg Ψ = deg Φ−j = np+p−1−j < np for j > p, and all coefficients of Ψ are divisible by j!. By Lemma 1.15 1 applied to Ψ(y , . . . , y ), we obtain j! 1 n anp n Ψ(β , . . . , β ) ∈ . j! 1 n Z Since j > p, anp n Ψ(β , . . . , β ) ∈ p , (p − 1)! 1 n Z § 1.3. Transcendentality of e and π 23 and it remains to note n X anp f (j)(β ) = n Ψ(β , . . . , β ), i (p − 1)! 1 n i=1 which proves (5).
Remark 1.3. Upon the conditions of Lemma 1.26, assume that βi ∈ Z, i = 1, . . . , n. Then the statement (5) of Lemma 1.26 may be enhanced in the obvious way as
j f (βi) ∈ pZ, i = 1, . . . , n, j > p. 2. Transcendentality of e.
Theorem 1.27. If α ∈ Q \{0} then eα is a transcendental number.
Proof. Suppose β = ea/b is an algebraic number for some a/b ∈ Q, a 6= 0. Then e is a root of the equation xa−βb = 0 with algebraic coefficients. By Theorem 1.13, all roots of such equation (in particular, e) are algebraic numbers. Hence, it is enough to show that e itself is transcendental. Assume e is algebraic, and let he(x) ∈ Q[x] be its minimal polynomial. Then there exists bn ∈ Z such that n bnhe(x) = b0 + b1x + ··· + bnx ∈ Z[x].
It is clear that b0 6= 0. Choose a prime number p ∈ Z such that p > n and p > |b0| (it is possible to make such a choice since the set of primes is infinite). Consider the polynomial
h(x) = (x − 1)(x − 2) ... (x − n) and construct 1 f(x) = H(h(x)) = xp−1h(x)p (p − 1)! as in (1.12), where βi = i, i = 1, . . . , n. For every k = 0, 1, . . . , n write the Hermite identity from Lemma 1.25:
k Z f(t)e−t dt = F (0) − F (k)e−k.
0 § 1.3. Transcendentality of e and π 24
k Multiply each of these equations by bke and add the results:
k n Z n X k−t X k −k bk f(t)e dt = bke (F (0) − F (k)e ) k=0 0 k=0 n n n X k X X = F (0) bke − bkF (k) = F (0)bnhe(e) − bkF (k). k=0 k=0 k=0 Therefore, k n Z n X k−t X bkf(t)e dt = − bkF (k). (1.13) k=0 0 k=0 Recall that X F (x) = f (j)(x). j>0 Consider the right-hand side of (1.13). Lemma 1.12 and Remark 1.3 imply
n n X (p−1) X (j) X X (j) bkF (k) = b0f (0) + b0 f (0) + bk f (k). k=0 j>p k=1 j>p (p−1) np p In the last expression, b0f (0) = (−1) b0(n!) 6≡ 0 (mod p) by the choice of p, all other summands are integer multiples of p. Hence, for every sufficiently large prime p the right-hand side of (1.13) is a nonzero integer number. Now, let us estimate the absolute value of the left-hand side of (1.13):
k n Z n X k−t X k−t bk f(t)e dt 6 |bk|k max {|f(t)e |} t∈[0,k] k=0 0 k=0 p n 1 p−1 np C1 6 (n+1) max |bk|n max {|f(t)|}e 6 C n (n−1) 6 C , k=0,...,n t∈[0,n] (p − 1)! (p − 1)! p C1 where C and C1 do not depend on the choice of p. Since lim = 0, p→∞ (p − 1)!
k n Z X k−t bk f(t)e dt < 1
k=0 0 § 1.3. Transcendentality of e and π 25 when p is sufficiently large, but the right-hand side of (1.13) is a nonzero integer and thus its absolute value is greater or equal to 1. The contradiction obtained proves the theorem. 3. Symmetrized n-tuples. N Definition 1.4. An N-tuple (β1, . . . , βN ) ∈ C is called symmetrized if N Y (x − βj) ∈ Q[x]. j=1 It is clear that a symmetrized tuple remains symmetrized after every permutation of its components. If we add (or remove) a rational number to (or from) a symmetrized tuple then the tuple obtained is symmetrized. Also, the concatenation of two or more symmetrized tuples is again a symmetrized tuple. To prove the transcendence of π we need the following properties of symmetrized tuples. n Lemma 1.28. Let (α1, . . . , αn) ∈ C be a symmetrized tuple, and let σ = σk ∈ Z[x1, . . . , xn], k ∈ {1, . . . , n}, be an elementary symmetric polynomial in x1, . . . , xn. Then N X n σ eα1 , . . . , eαn = eβj ,N = , k j=1 where (β1, . . . , βN ) is a symmetrized tuple. Proof. Recall that X σ = σk = xi1 . . . xik 16i1<···