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Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 A.J. Hildebrand Department of Mathematics University of Illinois http://www.math.uiuc.edu/~hildebr/ant Version 2013.01.07 2 Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 Contents Notation 8 0 Primes and the Fundamental Theorem of Arithmetic 11 0.1 Divisibility and primes . 11 0.2 The Fundamental Theorem of Arithmetic . 13 0.3 The infinitude of primes . 15 0.4 Exercises . 17 1 Arithmetic functions I: Elementary theory 19 1.1 Introduction and basic examples . 19 1.2 Additive and multiplicative functions . 21 1.3 The Moebius function . 24 1.4 The Euler phi (totient) function . 28 1.5 The von Mangoldt function . 29 1.6 The divisor and sum-of-divisors functions . 30 1.7 The Dirichlet product of arithmetic functions . 31 1.8 Exercises . 39 2 Arithmetic functions II: Asymptotic estimates 41 2.1 Big oh and small oh notations, asymptotic equivalence . 42 2.1.1 Basic definitions . 42 2.1.2 Extensions and remarks . 43 2.1.3 Examples . 44 2.1.4 The logarithmic integral . 48 2.2 Sums of smooth functions: Euler's summation formula . 50 2.2.1 Statement of the formula . 50 2.2.2 Partial sums of the harmonic series . 51 2.2.3 Partial sums of the logarithmic function and Stirling's formula . 52 3 4 CONTENTS 2.2.4 Integral representation of the Riemann zeta function . 55 2.3 Removing a smooth weight function from a sum: Summation by parts . 56 2.3.1 The summation by parts formula . 56 2.3.2 Kronecker's Lemma . 58 2.3.3 Relation between different notions of mean values of arithmetic functions . 60 2.3.4 Dirichlet series and summatory functions . 64 2.4 Approximating an arithmetic function by a simpler arithmetic function: The convolution method . 66 2.4.1 Description of the method . 66 2.4.2 Partial sums of the Euler phi function . 68 2.4.3 The number of squarefree integers below x . 70 2.4.4 Wintner's mean value theorem . 71 2.5 A special technique: The Dirichlet hyperbola method . 72 2.5.1 Sums of the divisor function . 72 2.5.2 Extensions and remarks . 74 2.6 Exercises . 76 3 Distribution of primes I: Elementary results 81 3.1 Chebyshev type estimates . 81 3.2 Mertens type estimates . 88 3.3 Elementary consequences of the PNT . 92 3.4 The PNT and averages of the Moebius function . 94 3.5 Exercises . 103 4 Arithmetic functions III: Dirichlet series and Euler prod- ucts 107 4.1 Introduction . 107 4.2 Algebraic properties of Dirichlet series . 108 4.3 Analytic properties of Dirichlet series . 115 4.4 Dirichlet series and summatory functions . 123 4.4.1 Mellin transform representation of Dirichlet series . 123 4.4.2 Analytic continuation of the Riemann zeta function . 126 4.4.3 Lower bounds for error terms in summatory functions 128 4.4.4 Evaluation of Mertens' constant . 130 4.5 Inversion formulas . 133 4.6 Exercises . 139 Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 INTRODUCTION TO ANALYTIC NUMBER THEORY 5 5 Distribution of primes II: Proof of the Prime Number The- orem 141 5.1 Introduction . 141 5.2 The Riemann zeta function, I: basic properties . 147 5.3 The Riemann zeta function, II: upper bounds . 148 5.4 The Riemann zeta function, III: lower bounds and zerofree region . 151 5.5 Proof of the Prime Number Theorem . 156 5.6 Consequences and remarks . 161 5.7 Further results . 164 5.8 Exercises . 169 6 Primes in arithmetic progressions: Dirichlet's Theorem 171 6.1 Introduction . 171 6.2 Dirichlet characters . 173 6.3 Dirichlet L-functions . 181 6.4 Proof of Dirichlet's Theorem . 182 6.5 The non-vanishing of L(1; χ) . 186 6.6 Exercises . 192 A Some results from analysis 195 P1 −2 A.1 Evaluation of n=1 n . 195 A.2 Infinite products . 196 Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 6 CONTENTS Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 List of Tables 1.1 Some important multiplicative functions . 23 1.2 Some other important arithmetic functions . 24 5.1 The error term in the Prime Number Theorem, I . 143 5.2 The error term in the Prime Number Theorem, II: Elementary proofs . 144 6.1 Table of all Dirichlet characters modulo 15. The integers a in the second row are the values of 2µ1 11µ2 modulo 15. 178 7 8 LIST OF TABLES Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 Notation R the set of real numbers C the set of complex numbers Z the set of integers N the set of positive integers (\natural numbers") N0 the set of nonnegative integers (i.e., N [ f0g) [x] the greatest integer ≤ x (floor function) fxg the fractional part of x, i.e., x − [x] d; n; m; : : : integers (usually positive) p; pi; q; qi;::: primes pm; pα;::: prime powers djn d divides n d - n d does not divide n m m m m+1 p jjn p divides exactly n (i.e., p jn and p - n) (a; b) the greatest common divisor (gcd) of a and b [a; b] the least common multiple (lcm) of a and b Qk αi n = i=1 pi canonical prime factorization of an integer n > 1 (with dis- tinct primes pi and exponents αi ≥ 1) Q α(p) n = p p the prime factorization of n in a different notation, with p running through all primes and exponents α(p) ≥ 0 Q m n = pmjjn p yet another way of writing the canonical prime factorization of n P summation over all primes ≤ x p≤x P summation over all prime powers pm with p prime and m a pm positive integer P summation over all positive divisors of n (including the trivial djn divisors d = 1 and d = n) P summation over all positive integers d for which d2 divides n d2jn P summation over all prime powers that divide exactly n (i.e., pmjjn Qk αi if n = i pi is the standard prime factorization of n, then k P m P αi f(p ) is the same as f(pi )) pmjjn i=1 P summation over all (distinct) primes dividing n. pjn 9 10 LIST OF TABLES Convention for empty sums and products: An empty sum (i.e., one in which the summation condition is never satisfied) is defined as 0; an empty Q m product is defined as 1. Thus, for example, the relation n = pmjjn p remains valid for n = 1 since the right-hand side is an empty product in this case. Math 531 Lecture Notes, Fall 2005 Version 2013.01.07 Chapter 0 Primes and the Fundamental Theorem of Arithmetic Primes constitute the holy grail of analytic number theory, and many of the famous theorems and problems in number theory are statements about primes. Analytic number theory provides some powerful tools to study prime numbers, and most of our current (still rather limited) knowledge of primes has been obtained using these tools. In this chapter, we give a precise definition of the concept of a prime, and we state the Fundamental Theorem of Arithmetic, which says that every integer greater than 1 has a unique (up to order) representation as a product of primes. We conclude the chapter by proving the infinitude of primes. The material presented in this chapter belongs to elementary (rather than analytic) number theory, but we include it here in order to make the course as self-contained as possible. 0.1 Divisibility and primes In order to define the concept of a prime, we first need to define the notion of divisibility. Given two integers d 6= 0 and n, we say that d divides n or n is divisible by d, if there exists an integer m such that n = dm. We write djn if d divides n, and d - n if d does not divide n. Note that divisibility by 0 is not defined, but the integer n in the above definition may be 0 (in which case n is divisible by any non-zero integer d) or negative (in which case djn is equivalent to dj(−n)). While the above definition allows for the number d in the relation \djn" 11 12 CHAPTER 0. PRIMES to be negative, it is clear that djn if and only if (−d)jn, so there is a one-to- one correspondence between the positive and negative divisors of an integer n. In particular, no information is lost by focusing on the positive divisors of a given integer, and it will be convenient to restrict the notion of a divisor to that of a positive divisor. We therefore make the following convention: Unless otherwise specified, by a divisor of an integer we mean a positive divisor, and in a notation like djn the variable d represents a positive divisor of n. This convention allows us, for example, to write the sum-of-divisors function σ(n) (defined as the sum of all positive divisors of n) simply as P σ(n) = djn d, without having to add the extra condition d > 0 under the summation symbol. The greatest common divisor (gcd) of two integers a and b that are not both zero is the unique integer d > 0 satisfying (i) dja and djb, and (ii) if cja and cjb, then cjd. The gcd of a and b is denoted by (a; b).
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