Notes for the course in Analytic Number Theory G. Molteni Fall 2019 revision 8.0 Disclaimer These are the notes I have written for the course in Analytical Number Theory in A.Y. 2011{'20. I wish to thank my former students (alphabetical order): Gu- glielmo Beretta, Alexey Beshenov, Alessandro Ghirardi, Davide Redaelli and Fe- derico Zerbini, for careful reading and suggestions improving these notes. I am the unique responsible for any remaining error in these notes. The image appearing on the cover shows a picture of the 1859 Riemann's scratch note where in 1932 C. Siegel recognized the celebrated Riemann{Siegel formula (an identity allowing to computed with extraordinary precision the values of the Riemann zeta function inside the critical strip). This image is a resized version of the image in H. M. Edwards Riemann's Zeta Function, Dover Publications, New York, 2001, page 156. The author has not been able to discover whether this image is covered by any Copyright and believes that it can appear here according to some fair use rule. He will remove it in case a Copyright infringement would be brought to his attention. Giuseppe Molteni This work is licensed under a Creative Commons Attribution-Non- Commercial-NoDerivatives 4.0 International License. This means that: (Attribu- tion) You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. (NonCommercial) You may not use the material for commercial purposes. (NoDerivatives) If you re- mix, transform, or build upon the material, you may not distribute the modified material. i Contents Disclaimer i Notation 1 Chapter 1. Prime number theorem 2 1.1. Preliminary facts: a warm-up 2 1.2. Two general formulas 5 1.3. The ring of arithmetical functions 13 1.4. Dirichlet series: as formal series 16 1.5. Dirichlet series: as complex functions 19 1.6. The analytic continuation of ζ(s) 28 1.7. Some elementary results 29 1.8. The Prime Number Theorem 39 Chapter 2. Primes in arithmetic progressions 61 Chapter 3. Sieve methods 71 3.1. Eratosthenes-Legendre's sieve 71 3.2. Selberg's Λ2-method 77 3.3. Sifting more classes 83 3.4. Two sets with positive density 91 Chapter 4. Sumsets 103 Chapter 5. Waring's problem 113 5.1. First step: cancellation in exponential sums 115 5.2. Second step: integral representation 120 Appendix. Bibliography 129 ii Notation • Let f and g : R ! [0; +1) functions. Then • f(x) = O(g(x)) as x ! x0 2 R := R [ {±∞} means that the quotient f(x)=g(x) is locally bounded in a neighborhood of x0, i.e. that there exist + a constant C 2 R and an open set U(x0) such that f(x)=g(x) ≤ C 8x 2 U(x0): • f(x) g(x) as x ! x0 2 R := R [ {±∞} is an equivalent notation for f(x) = O(g(x)). • f(x) = o(g(x)) as x ! x0 2 R := R [ {±∞} means that the quotient f(x)=g(x) tends to 0 as x ! x0 (in other words, the constant C in previous item can be taken arbitrarily small). • f(x) g(x) as x ! x0 2 R := R [ {±∞} means that both the quotients f(x)=g(x) and g(x)=f(x) are locally bounded in a neighborhood of x0, i.e. + that there exist a constant C 2 R and an open set U(x0) such that 1 g(x) ≤ f(x) ≤ Cg(x) 8x 2 U(x ): C 0 • f(x) = Ω(g(x)) as x ! x0 2 R := R [ {±∞} means that f(x) = O(g(x)) is false. This means that for every constant C 2 R and every open set U(x0) there exists x 2 U(x0) such that jf(x)j=jg(x)j > C: • Given x 2 R, the integer part of x is defined bxc := maxfn 2 Z: n ≤ xg. It must be not confused with dxe := minfn 2 Z: n ≥ xg. The fractional part of x is fxg := x − bxc (in signal processing this is called sawtooth func- tion). According to the definition, fxg is a 1-periodic function R ! R with discontinuities in every x 2 Z: lim fxg = 0; lim fxg = 1; 8n 2 Z: x!n+ x!n− • Usually we denote by s the complex argument of a complex function. In A. N. Th. it is customary to use σ and t to denote the real and the imaginary parts of s, respectively. In other words, s = σ + it with σ; t 2 R. • Given two integers m and n, mjn means that m divides n.(m; n) denotes their greatest common divisor, and [m; n] their smallest common multiple, so that mn = (m; n)[m; n]. • e(x) denotes the function e(x) := e2πix. 1 Chapter 1 Prime number theorem 1.1. Preliminary facts: a warm-up + Let P be the set of prime numbers. For every x 2 R , let π(x) := ]fp 2 P : p ≤ xg: How much large π(x) can be? Proposition 1.1 (Euclid) P is not finite, therefore π(x) ! 1 as x ! 1. Proof. Let p1; : : : ; pn be any set of primes. Let N := 1 + p1p2 ··· pn. N is an integer, thus it has a prime factor p. p is not equal to any pj, since N = 1 (mod pj). The argument can be modified in such a way to produce a quantitative result. Proposition 1.2 π(x) ln ln x as x ! 1. Proof. Let p1 = 2 < p2 < : : : be the complete set of primes (an infinite set, according to the previous result). We use the previous argument to prove that 2n−1 21−1 pn ≤ 2 for every n. In fact, the claim is true for n = 1 (because p1 = 2 ≤ 2 ). By induction on n and following the argument proving Proposition 1.1, we know that n−1 Y 2j−1 Pn−1 2j−1 2n−1−1 2n−1 pn ≤ 1 + p1p2 ··· pn−1 ≤ 1 + 2 = 1 + 2 j=1 = 1 + 2 ≤ 2 : j=1 For every x ≥ 2, let n be such that 22n−1 ≤ x < 22n . Then 2n−1 π(x) ≥ π(2 ) ≥ n ≥ log2 log2 x ln ln x: There are several alternative and elementary proofs of these facts. 2n P´olya. For every n 2 N let Fn := 2 +1, the nth Fermat number. These numbers are pairwise coprime, i.e. (Fn;Fm) = 1 8n 6= m: Proof. The sequence of Fermat numbers satisfies a kind of recursive formula; Qm−1 in fact k=0 Fj = Fm − 2 for every m ≥ 1, an identity which can be easily proved by induction. The formula shows that Fn divides Fm − 2 whenever n < m; in particular every prime dividing both Fn and Fm is also a factor of Fm and Fm − 2, hence it must be 2. Nevertheless, Fermat's numbers are odd, therefore they cannot have 2 as common factor. 2 CAP. 1: PRIME NUMBER THEOREM 3 The co-primality implies that every Fn has a special prime factor, pn, which divides Fn and does not divide every Fm with m 6= n. In particular, there are infinity many primes (because there are infinity many Fermat's numbers) and 2n−2 2n−1 the nth prime is lower than Fn−2 = 2 + 1 ≤ 2 as proved before. (Note that we use here the fact that p1 = 2 and p2 = 3 = F0). n P´olya (variation). For every n 2 N let Mn := 2 − 1, the nth Mersenne num- m−n ber. These numbers satisfy the relation Mm = 2 Mn + Mm−n for every m ≥ n, so that the greatest common divisor of Mm and Mn is M(m;n). Let p1 = 2; : : : ; pk be any set of distinct primes. Then Mp1 ;:::;Mpk are pairwise coprime, therefore there are at least k distinct odd prime numbers (because every Mn is an odd number), in particular there is at least one odd prime num- ber which is greater than pk, and hence one more prime. The argument proves pn that if 2 = p1 < p2 < p3 < ··· is the sequence of prime, then pn+1 < 2 . This upper bound for pn can be used to produce a lower bound for π(x), but of incredibly low quality. Erd}os. Every integer n can be written in a unique way as a product of a square 2 m and a squarefree q. Fixp x > 2 and apply that decomposition to every integer n ≤ x. There are x possible values for m, and 2π(x) values for q, at most. Hence p π(x) bxc = ]fn 2 N: n ≤ xg ≤ ]fmg · ]fqg ≤ x · 2 implying that π(x) ln x. Note that also that this simple argument already improves Proposition 1.2. Euler. Consider the product Y Y 1 1 1 (1 − 1=p)−1 = 1 + + + + ··· : p p2 p3 p≤x p≤x Every integer n can be written in a unique way as product of prime powers and when n is ≤ x then also the primes appearing in its factorization are ≤ x (trivial). Therefore the previous product gives Y Y 1 1 1 X 1 (1.1) (1 − 1=p)−1 = 1 + + + + ··· ≥ : p p2 p3 n p≤x p≤x n≤x The right hand side diverges in x, hence this inequality already proves the existence of infinitely many primes: only in this case the product appearing to the left hand side diverges.