Analytic Number Theory

Transcription

1 Analytic Number Theory

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3 Analytic Number Theory Travis Dirle December 4, 2016

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5 Contents 1 Summation Techniques Abel Summation Euler-Maclaurin Summation Arithmetic Functions Arithmetic Functions The Ring of Arithmetic Functions Important Relationships Dirichlet Series Analytic Properties of Dirichlet Series Euler Products Summation Formulae Average and Extremal Orders Average Orders Extremal Orders Sieve Methods The Sieve of Eratosthenes Brun s Combinatorial Sieve and Twin Primes Characters Dirichlet Characters Primitive Characters Gauss Sums Quadratic Reciprocity Legendre s Symbol and its Properties The Jacobi Symbol The Functions ζ(s) and L(s, χ) The Zeta Function Approximations and Bounds i

6 CONTENTS 8.3 Dirichlet L-functions The Prime Number Theorem Theorems of Chebyshev and Mertens The Prime Number Theorem ii

7 Chapter 1 Summation Techniques 1.1 ABEL SUMMATION Classically one calls Abel summation (or also partial summation) the process whereby one transforms a finite sum of products of two terms by means of the partial sums of one of them. Thus, by letting A 0 = 0, we have A n = n m=1 a m(n 1), N a n b n = N (A n A n 1 )b n = = N 1 N A n b n A n (b n b n+1 ) + A N b N. N 1 A n b n+1 Theorem (Abel s identity) For any arithmetic function a(n) let A(x) = n x a(n), where A(x) = 0 if x < 1. Assume f has a continuous derivative on the interval [y, x], where 0 < y < x. Then we have a(n)f(n) = A(x)f(x) A(y)f(y) x y<n x y A(t)f (t) dt. Theorem Let {a n } be a sequence of complex numbers. Set A(t) = a n (t > 0). n t Let b(t) be a continuously differentiable function on the interval [1, x]. Then we have x a n b(n) = A(x)b(x) A(t)b (t) dt. 1 n x 1 1

8 CHAPTER 1. SUMMATION TECHNIQUES Theorem (Comparison of a sum and an integral) Let f be a real monotonic function on the interval [a, b], with a, b Z. Then there exists some real number θ = θ(a, b), 0 θ 1, such that b f(n) = f(t) dt + θ(f(b) f(a)) a<n b a Theorem (Second mean value formula) Let f(t) be monotonic and g(t) be integrable on the real interval [a, b]. Then there exists a real number ξ, with a ξ b, such that b a f(t)g(t) dt = f(a) ξ a g(t) dt + f(b) b ξ g(t) dt. 1.2 EULER-MACLAURIN SUMMATION Definition For any complex x we define the functions B n (x) by the equation ze xz e z 1 = B n (x) z n, where z < 2π. n! n=0 The numbers B n (0) are called Bernoulli numbers and are denoted by B n. Thus, z e z 1 = n=0 B n n! zn, where z < 2π. Theorem The functions B n (x) are polynomials in x given by n ( ) n B n (x) = B k x n k. k k=0 Theorem (Euler-Maclaurin summation formula) For any integer k 0 and for any function f of class C k+1 on [a, b], a, b Z, we have b k ( 1) r+1 B r+1 f(n) = f(t) dt + (f (r) (b) f (r) (a)) (r + 1)! a<n b a + ( 1)k (k + 1)! r=0 b a B k+1 (t)f (k+1) (t) dt. Theorem (Euler s summation formula) If f has a continuous derivative f on the interval [y, x], where 0 < y < x, then x x f(n) = f(t) dt + (t [t])f (t) dt y<n x y +f(x)([x] x) f(y)([y] y). y 2

9 Chapter 2 Arithmetic Functions 2.1 ARITHMETIC FUNCTIONS Definition A real or complex valued function defined on the positive integers is called an arithmetical function or a number-theoretic function. Definition A function is called additive if it satisfies f(mn) = f(m) + f(n) whenever (m, n) = 1; it is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1; it is called completely additive or multiplicative if the prior respective condition holds even when (m, n) 1, that is if f(p v ) = vf(p) or f(p v ) = f(p) v respectively. One says that f is strongly additive or multiplicative if, besides prior conditions, f(p v ) = f(p) for all v 1. Some notation: p v a means that p v a but p v+1 a. The main interest in these notions is that an additive or multiplicative function respects the multiplicative structure of N in the sense that the image of an integer is the sum or product of the images of the prime powers arising in its canonical factorisation. We can thus write f(n) = p v n f(p v ), and f(n) = p v n f(p v ), when f is respectively additive or multiplicative. Some classical examples of arithmetic functions: The counting functions of the total number of prime factors of n, taken with or without multiplicity, respectively: Ω(n) = p v n v and ω(n) = p v n 1 = p n 1. 3

10 CHAPTER 2. ARITHMETIC FUNCTIONS The divisor function and the sum of kth powers of divisors, denoted respectively by τ(n) = d n 1 and σ k (n) = d n d k (k C). We have that τ(n) = σ 0 and is sometimes denoted d(n). When k = 1, σ 1 (n) is the sum of the divisors, and is often denoted by σ(n). This function is multiplicative. Euler s totient function is defined to be the number of positive integers not exceeding n which are relatively prime to n φ(n) = 1. 1 h n,(h,n)=1 The Möbius function, defined by { ( 1) ω(n), if n is squarefree, µ(n) = 0, otherwise. The Von Mangoldt s function { log p, if n = p v for some v 1, Λ(n) = 0, if n is not a prime power. The identity function given by 1 δ(n) = = n { 1 if n = 1, 0 if n > 1. The unit function denoted u or 1 is the arithemtic function such that u(n) = 1 for all n. Liouville s function λ is a completely multiplicative function defined by λ(n) = ( 1) Ω(n) Theorem The divisor function is multiplicative. We have τ(n) = p v n(v + 1), for (n 1). The Möbius function is multiplicative. We have µ(n) = p v n µ(p v ). Theorem We have that Ω and ω are additive, the former completely, the latter strongly. 4

11 CHAPTER 2. ARITHMETIC FUNCTIONS 2.2 THE RING OF ARITHMETIC FUNCTIONS Definition Let f be an arithmetic function. The formal Dirichlet series associated with f is the formal series D(f; s) = f(n)n s. The sum and product of two formal Dirichlet series are defined in a natural way by D(f; s) + D(g; s) = (f(n) + g(n))n s, with D(f; s)d(g; s) = h(n) = dd =n h(n)n s, f(d)g(d ). The set of formal Dirichlet series equipped with these two operations has the structure of a commutative ring with unity given by the series D(δ; s) = 1 The correspondence between arithmetic functions and formal Dirichlet series induces on the set of arithmetic functions an addition + and a product for which the fundamental properties are respectively D(f + g; s) = D(f; s) + D(g; s) D(f g; s) = D(f; s)d(g; s). We thus have (f + g)(n) = f(n) + g(n) and (f g)(n) = h(n) where h is as defined above. Definition The product is called Dirichlet convolution. Definition A necessary and sufficient conditon for an arithmetic f to be invertible is that f(1) 0. Under this assumption the family of equations f(n/d)g(d) = δ(n) for (n 1) d n allows us to calculate the inverse function g(n) recursively: { g(1) = f(1) 1 g(n) = f(1) 1 d n,d<n f(n/d)g(d) for (n > 1). 5

12 CHAPTER 2. ARITHMETIC FUNCTIONS Theorem The group G of units in the ring A of arithmetic functions consists of the functions f such that f(1) 0. Theorem A necessary and sufficient condition for a function f A to be multiplicative is that its associated formal series D(f; s) be expandable as an infinite formal product of Eulerian type, that is D(f; s) = p (1 + f(p v )p vs ). v=1 Theorem The set M of multiplicative arithmetic functions is a subgroup of the group G of units of A. Note that the Dirichlet product of two completely multiplicative functions need not be completely multiplicative. Theorem Let f be multiplicative. Then f is completely multiplicative if and only if f 1 (n) = µ(n)f(n) for all n 1. Theorem If f is multiplicative, we have µ(d)f(d) = (1 f(p)). p n d n Theorem (First Möbius inversion formula) Let f and g be arithmetic functions. The two following properties are equivalent (i) g(n) = d n f(d) for (n 1) (ii) f(n) = d n g(d)µ(n/d) for (n 1). The Möbius inversion is really saying g = f 1 f = g µ. Theorem (Second Möbius inversion formula) Let F and G be functions defined on [1, ]. The following conditions are equivalent (i) F (x) = n x G(x/n) for (x 1) (ii) G(x) = n x µ(n)f (x/n) for (x 1). 6

13 CHAPTER 2. ARITHMETIC FUNCTIONS Theorem We have that 2.3 IMPORTANT RELATIONSHIPS τ(n) = d n 1 = d n 1(d)1(n/d), from which τ = 1 1. Theorem Let j denote the identity function j(n) = n for (n 1). Then clearly we have σ = 1 j and also σ k (n) = d n d k = (1 j k )(n) for all real or complex values of the parameter k. Theorem The Möbius function is the convolution inverse of the function 1, that is 1 µ = δ. In other words, µ(d) = d n { 1 (n = 1) 0 (n > 1). Theorem Von Mangoldt s function can be defined as Λ = µ log or also, Λ = µ log 1. Theorem If n 1 we have log n = d n Λ(d). Definition (Chebyshev s summatory functions) ψ(x) = n x Λ(n) = log lcm{n : n x}, θ(x) = p x log p 7

14 CHAPTER 2. ARITHMETIC FUNCTIONS Theorem For x 1 we have the representation ψ(x) = θ(x 1/k ) Definition We denote by π(x) the number of primes not exceeding x. Theorem For x 2, we have k=1 ψ(x) = θ(x) + O( x), π(x) = θ(x) ( ) x log x + O. (log x) 2 Corollary Let a, b, be constants such that 0 < a < log 2, b > log 4. For all sufficiently large x, we have ax θ(x) ψ(x) bx. Theorem Euler s totient function has the following properties: (i) φ(p a ) = p a p a 1 (ii) φ(mn) = φ(m)φ(n)(d/φ(d)), where d = (m, n) (iii) φ(mn) = φ(m)φ(n) if (m, n) = 1 (iv) a b φ(a) φ(b) (v) φ(n) is even for n 3. Moreover, if n has r distinct odd prime factors, then 2 r φ(n). Theorem For each n 1 we can write the Euler totient function as follows φ(n) = δ ((m, n)), from which we infer that 1 m n φ = µ j. In particular for each prime p we have ( φ(p v ) = µ(1)p v + µ(p)p v 1 = p v 1 1 ). p Theorem Euler s totient function is multiplicative. For any integer n 1, we have φ(n) = n ( 1 1 ). p p n Theorem We also have that n = d n φ(d). In other words, j = 1 φ. 8

15 CHAPTER 2. ARITHMETIC FUNCTIONS Theorem For every n 1 we have { 1 if n is a square, λ(d) = 0 otherwise. d n Also, λ 1 (n) = µ(n) for all n. 9

16 CHAPTER 2. ARITHMETIC FUNCTIONS 10

17 Chapter 3 Dirichlet Series 3.1 ANALYTIC PROPERTIES OF DIRICHLET SERIES Definition A Dirichlet series is a series of the form α(s) = a nn s where s is a complex variable. Theorem If β(s) = m=1 b mm s is a second Dirichlet series and γ(s) = α(s)β(s), then we have that ( ) γ(s) = a k k s b m m s = a k b m (km) s = a k b m n s. k=1 m=1 k=1 m=1 km=n That is, γ(s) is a Dirichlet series, γ(s) = c nn s, whose coefficients are c n = a k b m. km=n The mere convergence of α(s) and β(s) is not sufficient to justify this. It is justified if the series are absolutely convergent. Definition Among the Dirichlet series we shall consider is the Riemann zeta function which for σ > 1 is defined by the absolutely convergent series ζ(s) = n s. Definition We say that a function f(x) is asymptotic to g(x) as x tends to some limiting value (say x ), and write f(x) g(x), if f(x) lim x g(x) = 1. Definition We say that f(x) is small oh of g(x) and write f(x) = o(g(x)), if f(x)/g(x) 0 as x tends to its limit. 11

18 CHAPTER 3. DIRICHLET SERIES Definition We say that f(x) is big oh of g(x) and write f(x) = O(g(x)) if there is a constant C > 0 such that f(x) Cg(x) for all x in the appropriate domain. The function f may be complex, but g is necessarily non-negative. We may also write f(x) g(x). Definition If both f g and g f then we say that f and g have the same order of magnitude, and write f g. Theorem Suppose that the Dirichlet series α(s) = a nn s converges at the point s = s 0, and H > 0 is an arbitrary constant. Then the series α(s) is uniformly convergent in the sector S = {s : σ σ 0, t t 0 H(σ σ 0 )}. Definition Any Dirichlet series α(s) = a nn s has an abscissa of convergence σ c with the property that α(s) converges for all s with σ > σ c, and for no s with σ < σ c. Moreover, if s 0 is a point with σ 0 > σ c, then there is a neighbourhood of s 0 in which α(s) converges uniformly. Theorem Let A(x) = n x a n. If σ c < 0, then A(x) is a bounded function, and a n n s = s A(x)x s 1 dx for σ > 0. If σ c 0, then lim sup x and prior sum holds for σ > σ c. 1 log A(x) log x = σ c, Definition In general we let σ a denote the infimum of those σ for which a n n σ <. Then σ a, the abscissa of absolute convergence, is the abscissa of convergence of the series a n n s, and we see that a n n s is absolutely convergent if σ > σ a, but not if σ < σ a. The strip of conditional convergence is σ c σ σ a. Theorem In the above notation, we have σ c σ a σ c + 1. Remark We have an interest in Dirichlet series expressed in terms of the size of t. Our interest is in large values of this quantity, but in order that the statements be valid for small t we sometimes write t +4, or just let τ = t +4. Theorem Suppose that α(s) = a n n s has abscissa of convergence σ c. If δ and ɛ are fixed, 0 < ɛ < δ < 1, then α(s) τ 1 δ+ɛ uniformly for σ σ c + δ. The implicit constant may depend on the coefficients a n, on δ, and on ɛ. 12

19 CHAPTER 3. DIRICHLET SERIES Theorem If a n n s = b n n s for all s with σ > σ 0 then a n = b n for all positive integers n. Theorem (Landau) Let α(s) = a n n s be a Dirichlet series whose abscissa of convergence σ c is finite. If a n 0 for all n then the point σ c is a singularity of the function α(s). 3.2 EULER PRODUCTS Theorem Let α(s) = a n n s and β(s) = b n n s be two Dirichlet series, and put γ(s) as their product. If s is a point at which the two series α(s) and β(s) are both absolutely convergent, then γ(s) is absolutely convergent and γ(s) = α(s)β(s). If f is multiplicative then the Dirichlet series f(n)n s factors into a product over primes. To see why, we first argue formally (ignoring questions of convergence). When the product (1 + f(p)p s + f(p 2 )p 2s + f(p 3 )p 3s + ) is expanded, the generic term is p f(p k 1 1 )f(p k 2 2 ) f(p kr r ) (p k 1 1 p k 2 2 p kr r ) s Set n = p k 1 1 p k 2 2 p kr r. Since f is multiplicative, the above is f(n)n s. Moreover, this correspondence between products of prime powers and positive integers n is one-to-one, in view of the fundamental theorem of arithmetic. Hence after rearranging the terms, we obtain the sum f(n)n s. That is f(n)n s = p (1 + f(p)p s + f(p 2 )p 2s + ). Definition The product on the right-hand side is called the Euler product of the Dirichlet series. Theorem If f is multiplicative and f(n) n σ <, then f(n)n s can be expressed as its Euler product. 13

20 CHAPTER 3. DIRICHLET SERIES Let 3.3 SUMMATION FORMULAE F (s) = a n n s be a Dirichlet series with abscissa of convergence σ c and abscissa of absolute convergence σ a. Extending the definition of the function n a n by setting a x = 0 if x R\N, we introduce the normalized summatory function A (x) = n<x a n a x for x 0. Theorem (Perron s formula) Let k > max(0, σ c ). We have A (x) = 1 2πi k+i k i F (s)x s s 1 ds for x > 0, where the integral is conditionally convergent for x R\N and converges in the sense of Cauchy s principal value when x N. Theorem (First effective Perron formula) For k > max(0, σ a ), T 1 and x 1, we have A(x) = 1 ( k+it F (s)x s ds ) 2πi k it s + O x k a n. n k (1 + T log(x/n) ) Corollary (Second effective Perron formula) Let F (s) = a nn s be a Dirichlet series with finite abscissa of absolute convergence σ a. Suppose that there exists some real number α 0 such that (i) a n n σ (σ σ a ) α for (σ > σ a ), and that B is a non-decreasing function satisfying (ii) a n B(n) for n 1. Then for x 2, T 2, σ σ a, k = σ a σ + 1/ log x, we have n x a n n s = 1 2πi k+it k it σa σ (log x)α +O (x + B(2x) T x σ F (s + w)x w dw w ( 1 + x log T )). T Theorem For k > max(0, σ c ) and x 1, we have 14 n x a n log(x/n) = 1 2πi k+i k i F (s)x s ds s 2,

21 CHAPTER 3. DIRICHLET SERIES and x 0 A(t) dt = 1 2πi k+i k i F (s)x s+1 ds s(s + 1). 15

22 CHAPTER 3. DIRICHLET SERIES 16

23 Chapter 4 Average and Extremal Orders 4.1 AVERAGE ORDERS Definition Euler s constant γ is defined by ( n ) 1 γ = lim n k log n. Or also We have γ γ = 1 k=1 1 t [t] t 2 Definition We say that an arithmetic function F (n) has a mean value c if lim N 1 N dt N F (n) = c. Definition We define an average order of an arithmetic function f to be any elementary function g of a real variable such that g(n). n x f(n) n x Theorem (Dirichlet s hyperbola method) Let f, g be two arithmetic functions, with respective summatory functions F,G. For 1 y x, we have g)(n) = n x(f g(n)f (x/n) + f(m)g(x/m) F (x/y)g(y). n y m x/y Theorem As x tends to infinity, we have τ(n) = x(log x + 2γ 1) + O( x) n x where γ denotes Euler s constant. 17

24 CHAPTER 4. AVERAGE AND EXTREMAL ORDERS Theorem As x tends to infinity, we have n x σ(n) = π2 12 x2 + O(x log x) Theorem As x tends to infinity, we have φ(n) = 3 π 2 x2 + O(x log x). n x Theorem As x tends to infinity, we have ω(n) = x log 2 x + c 1 x + O n x where c Theorem As x tends to infinity, we have Ω(n) = x log 2 x + c 2 x + O n x ( ) x. log x ( x ) log x with c 2 = c 1 + p 1 p(p 1) Theorem The following statements are equivalent: (i) ψ(x) x as x, (ii) M(x) = n x µ(n) = o(x) as x, (iii) µ(n)/n = 0. (iv) π(x) x/ log x as x, (v) θ(x) x as x, (vi) n x Λ(n)/n = log x γ + o(1) as x. Theorem As x tends to infinity, we have µ(n) 2 = 6 π x + O( x). 2 n x 18

25 CHAPTER 4. AVERAGE AND EXTREMAL ORDERS 4.2 EXTREMAL ORDERS Although the average order of an arithmetic function gives an intuitive idea of its overall behavior, it can only reflect crudely the variations of its values. Here, we describe standard methods leading to individual bounds that are optimal. Let f be an arithmetic function. Let g, h denote elementary monotone functions such as logarithms or powers. The information concerning f may be one of the following types: (i) f(n) = O(g(n)) for (n n 0 ) (ii) f(n) = o(g(n)) for (n ) (iii) f(n) = (1 + o(1))g(n) for (n ) (iv) f(n) = g(n) + O(h(n)) for (n n 0 ). In the last case it is desirable that h(n) be o(g(n)). A measure of optimality of these asymptotic relations can be expressed by estimates of the following form (where g(n) is assumed to be ultimately positive): (a) f(n) = Ω + (g(n)) [i.e. lim sup f(n)/g(n) > 0] (b) f(n) = Ω (g(n)) [i.e. lim inf f(n)/g(n) < 0] (c) f(n) = Ω(g(n)) [i.e. lim sup f(n) /g(n) > 0]. Definition Let f be an arithmetic function and let g be a non-decreasing function which is ultimately positive. We say that g is a maximal order (resp. minimal order) for f if lim sup f(n)/g(n) = 1 (resp. lim inf f(n)/g(n) = 1). n n Theorem Let f be a multiplicative function. If lim p v f(p v ) = 0, then lim n f(n) = 0. Corollary For all ɛ > 0 we have τ(n) = O ɛ (n ɛ ). Theorem A maximal order for the function log τ(n) is log 2 log n/ log 2 n. Theorem (i) A maximal order for the function ω(n) is log n/ log 2 n. (ii) A maximal order for the function Ω(n) is log n/ log 2. Theorem A maximal order for φ(n) is n. A minimal order is where γ denotes Euler s constant. e γ n/ log 2 n 19

26 CHAPTER 4. AVERAGE AND EXTREMAL ORDERS Theorem For K > 0 a minimal order for σ K (n) is n K. For K > 1, a maximal order is ζ(k)n K. For K = 1, a maximal order is e γ n log 2 n. For 0 < K < 1, we have { } σ K (n) n K (log n) 1 K exp (1 + o(1)) (1 K) log 2 n and the opposite inequality is satisfied when n runs through a suitable infinite sequence of integers. 20

27 Chapter 5 Sieve Methods 5.1 THE SIEVE OF ERATOSTHENES The sieve of Eratosthenes is a method for counting the number N m (x) of positive integers n not exceeding x that are relatively prime to a fixed number m. Theorem (The sieve of Eratosthenes) We have that N m (x) = d m ( 1) ω(d) [x/d], where m denotes the square free kernel of m, i.e. m denotes the number of distinct prime factors of d. Theorem (Legendre s formula) We have that = k j=1 p j, and ω(d) N m (x) = d m µ(d)[x/d] Theorem When we select m = p x p the only numbers counted by N m (x) are 1 and the prime numbers p from the interval [ x, x]. Thus N m (x) = π(x) π( x) + 1, and we obtain π(x) = 1 + π( x) + µ(d)[x/d]. P + (d) x 5.2 BRUN S COMBINATORIAL SIEVE AND TWIN PRIMES Theorem (Brun) Denote by χ t the characteristic function of the set of integers n such that ω(n) t. Then for each integer h 0 the functions defined by µ i (n) = µ(n)χ 2h+2 i (n) satisfy the inequalities µ 1 1 δ µ

28 CHAPTER 5. SIEVE METHODS In other words, we have µ 1 (d)[x/d] N m (x) µ 2 (d)[x/d] d m d m Corollary Let A be a finite set of integers and let P be a set of prime numbers. Write A d = {a A : a 0 mod d}, P (y) = p, p P,p y S(A, P, y) = {a A : (a, P (y)) = 1}. Then for each integer h 0 we have µ(d)a d S(A, P, y) d P (y),ω(d) 2h+1 d P (y),ω(d) 2h µ(d)a d. Theorem (Brun-Titchmarsh) Let us write π(x; l, k) = {p x : p l mod k}. Let x, y be positive integers and k, l be integers. If y/k, we have y π(x + y; l, k) π(x; l, k) (2 + o(1)) φ(k) log(y/k). Define the following sets as follows J = {p : p + 2 is prime } and J(x) = J [1, x]. Definition The twin prime constant is defined as follows C = 2 p 3(1 (p 1) 2 ). Theorem As x tends to infinity, we have J(x) (8C + o(1))x/(log x) 2. Corollary We have p J 1 p <. 22

29 Chapter 6 Characters 6.1 DIRICHLET CHARACTERS Definition Let G be an arbitrary group. A complex-valued function χ defined on G is called a character of G if χ has the property χ(ab) = χ(a)χ(b) for all a, b G, and if χ(c) 0 for some c G. Theorem If χ is a character of a finite group G with identity element e, then χ(e) = 1 and each function value χ(a) is a root of unity. In fact, if a n = e then χ(a) n = 1. Definition Every group G has at least one character, namely the function which is identically 1 on G. This is called the principal character. Theorem A finite abelian group G of order n has exactly n distinct characters. For the remainder of time, G is a finite abelian group of order n. The principal character will be denoted χ 0. The other characters have the property that χ(a) 1 for some a G. We use e(θ) to denote e 2πiθ. Theorem If multiplication of characters is defined by the relation (χ i χ j )(a) = χ i (a)χ j (a) for each a in G, then the set of characters of G forms an abelian group of order n. We denote this group by Ĝ. The identity element of Ĝ is the principal character χ 0. The inverse of χ i is the reciprocal 1/χ i. Remark For each character χ we have χ(a) = 1. Hence the reciprocal 1/χ(a) is equal to the complex conjugate χ(a). Thus the function χ defined by χ(a) = χ(a) is also a character of G. Moreover, we have χ(a) = 1 χ(a) = χ(a 1 ) 23

30 CHAPTER 6. CHARACTERS for every a G. Lemma Suppose that G is cyclic of order n, say G = (a). Then there are exactly n characters of G, namely χ k (a m ) = e(km/n) for 1 k n. Moreover, { n if χ = χ 0, χ(g) = 0 otherwise, g G and χ(g) = χ Ĝ { n if g = e, 0 otherwise. In this situation, Ĝ is cyclic, Ĝ = (χ 1). Theorem Let G be a finite abelian group. Then Ĝ = G, and the previous two properties hold. Definition A Dirichlet character is a multiplicative homomorphism of the group (Z/qZ) into the complex numbers of modulus 1. We extend such a character to N by setting χ(n) = χ(a) if n a mod q, (n, q) = 1, and χ(n) = 0 if (n, q) > 1. Corollary The multiplicative group (Z/qZ) of reduced residue classes mod q has φ(q) Dirichlet characters. If χ is such a character, then { φ(q) if χ = χ 0, χ(n) = 0 otherwise.,(n,q)=1 If (n, q) = 1, then χ(n) = χ { φ(q) if n 1 mod q, 0 otherwise, where the sum is extended over the φ(q) Dirichlet characters χ mod q. Theorem Each Dirichlet character modulo q is completely multiplicative with period q. That is, we have and χ(mn) = χ(m)χ(n) for all m,n χ(n + q) = χ(n) for all n Conversely, if χ is completely multiplicative and periodic with period q, and if χ(n) = 0 if (n, q) > 1, then χ is one of the Dirichlet characters modulo q. 24

31 CHAPTER 6. CHARACTERS Corollary If χ i is a character mod q i for i = 1, 2, then χ 1 (n)χ 2 (n) is a character mod [q 1, q 2 ]. If q = q 1 q 2, (q 1, q 2 ) = 1, and χ is a character mod q, then there exist unique characters χ i mod q, i = 1, 2, such that χ(n) = χ 1 (n)χ 2 (n) for all n. Theorem (Orthogonality relations) (i) For all integers n, m 1, we have { φ(q) 1 1 if n m mod q and (m, q) = 1, χ(n)χ(m) = 0 otherwise. χ mod q (ii) For all Dirichlet characters χ, χ, to the modulus q, we have φ(q) { 1 1 if χ = χ, χ(n)χ (n) = 0 otherwise. 1 n q 6.2 PRIMITIVE CHARACTERS Definition Let χ be a Dirichlet character modulo k and let d be any positive divisor of k. The number d is called an induced modulus for χ if we have χ(a) = 1 whenever (a, k) = 1 and a 1 mod d. Definition Suppose that d q and that χ is a character modulo d, and set { χ (n) (n, q) = 1 χ(n) = 0 otherwise. Then χ(n) is a Dirichlet character modulo q. In this situation we say that χ induces χ. Theorem Let χ be a Dirichlet character modulo k. Then 1 is an induced modulus for χ if and only if χ = χ 0. Definition A Dirichlet character χ mod k is said to be primitive modulo k if it has no induced modulus d < k. In other words, χ is primitive modulo k if and only if for every divisor d of k, 0 < d < k, there exists an integer a 1 mod d, (a, k) = 1, such that χ(a) 1. Theorem Every nonprincipal character χ modulo a prime p is a primitive character modulo p. Theorem Let χ be a Dirichlet character modulo k and assume d k, d > 0. Then d is an induced modulus for χ if and only if χ(a) = χ(b) whenever (a, k) = (b, k) = 1 and a b mod d. 25

32 CHAPTER 6. CHARACTERS Definition Let χ be a Dirichlet character modulo k. The smallest induced modulus d for χ is called the conductor of χ. Theorem Let χ denote a Dirichlet character modulo q and let d be the conductor of χ. Then d q, and there is a unique primitive character χ modulo d that induces χ. 6.3 GAUSS SUMS Definition Given a character χ modulo q, we define the Gauss sum τ(χ) of χ to be q τ(χ) = χ(a)e(a/q). a=1 This may be regarded as the inner product of the multiplicative character χ(a) with the additive character e(a/q). The Gauss sum is a special case of the more general sum q c χ (n) = χ(a)e(an/q) a=1 When χ is the principal character, this is Ramanujan s sum q c q (n) = e(an/q). a=1,(a,q)=1 Theorem Suppose that χ is a character modulo q. If (n, q) = 1, then q χ(n)τ(χ) = χ(a)e(an/q), and in particular a=1 τ(χ) = χ( 1)τ(χ). Theorem Suppose that (q 1, q 2 ) = 1, that χ i is a character modulo q i for i = 1, 2, and that χ = χ 1 χ 2. Then τ(χ) = τ(χ 1 )τ(χ 2 )χ 1 (q 2 )χ 2 (q 1 ). Theorem Suppose that χ is a primitive character modulo q. Then previous theorem holds for all n, and τ(χ) = q. Corollary Suppose that χ is a primitive character modulo q. Then for any integer n, χ(n) = 1 q χ(a)e(an/q). τ(χ) 26 a=1

33 CHAPTER 6. CHARACTERS Theorem Suppose that χ is a primitive character modulo q with q > 1. If χ( 1) = 1, then while if χ( 1) = 1, then L(1, χ) = τ(χ) q q 1 χ(a) log(sin πa/q), a=1 L(1, χ) = iπτ(χ) q 2 q 1 aχ(a). Theorem Let χ be a character modulo q that is induced by the primitive character χ modulo d. Then τ(χ) = µ(q/d)χ (q/d)τ(χ ). a=1 27

34 CHAPTER 6. CHARACTERS 28

35 Chapter 7 Quadratic Reciprocity 7.1 LEGENDRE S SYMBOL AND ITS PROPERTIES Definition Given an odd prime p and an integer d relatively prime to p, we define the Legendre symbol as follows: ( ) d = +1, if d is a quadratic residue mod p, p ( ) = 1, if d is a non-residue mod p d p The multiplicative group F p being cyclic of even order p 1, the squares in F p form a subgroup (F p) 2 of index 2 and F p/(f p) 2 is isomorphic to {+1, 1}. The Legendre symbol stands for the compositon of the following homomorphisms: Z pz F p F p/(f p) 2 = {+1, 1}. As a consequence there is the formula: ( ) ( ) ( ) ab a b =, with a, b Z pz. p p p Proposition (Euler s Criterion) If p is an odd prime and if p a, then ( ) a a (p 1)/2 mod p. p Theorem (The Law of Quadratic Reciprocity) If p and q are distinct odd prime numbers, then ( ) ( ) p q = ( 1) (p 1)(q 1)/4. q p Proposition ( ) (The Complementary Formulas) If p is an odd prime, then 1 (i) p ( ) = ( 1) (p 1)/2 and (ii) = ( 1) (p2 1)/8. 2 p 29

36 CHAPTER 7. QUADRATIC RECIPROCITY Proposition (The First Supplementary Law) Since (p 1)/2 is even if p 1 mod 4 and odd if p 3 mod 4, then we have ( ) { 1 = ( 1) (p 1)/2 1 if p 1 mod 4 = p 1 if p 3 mod 4. Proposition (The Second Supplementary Law) ( ) 2 = ( 1) (p2 1)/8 p so ( ) { 2 1 if p ±1 mod 8 = p 1 if p ±3 mod 8. Proposition (Fermat) Any prime number p 1 mod 4 may be represented as the sum of two squares. Theorem (Lagrange) Every natural number may be represented as the sum of four squares. Hence any prime number is the sum of four squares. Theorem If g is a primitive root mod p, then g is a quadratic nonresidue mod p. Consequently, exactly half of the integers 1, 2,..., p 1 are quadratic residues and half quadratic nonresidues. Definition Let K be a field of characteristic p such that K contains the pth roots of unity. Let ζ K be a primitive pth root of unity. Define the Gauss sum by p 1 ( ) a τ p = ζ a. p a=1 Theorem We have that τ 2 p = ( 1) (p 1)/2 p. 7.2 THE JACOBI SYMBOL Definition Let Q be an odd positive integer with prime factorization Q = q 1 q s. The Jacobi symbol is defined by ( ) a s ( ) a =. Q 30 It follows directly from the definition that ( ) ( ) ( ) ( ) ( ) ( ) a a a a a aa =, =. Q Q QQ Q Q Q i=1 q i

37 CHAPTER 7. QUADRATIC RECIPROCITY Also a a mod Q ( ) a = Q ( ) a because if a a mod Q then a a mod q i for i = 1,..., s. Theorem If Q is an odd positive integer then ( ) 1 = ( 1) (Q 1)/2 Q Theorem If Q is an odd positive integer, then ( ) 2 = ( 1) (Q2 1)/8 Q Theorem If P and Q are odd, relatively prime positive integers, then ( ) ( ) P Q (P 1)(Q 1)/4 = ( 1) Q P Q 31

38 CHAPTER 7. QUADRATIC RECIPROCITY 32

39 Chapter 8 The Functions ζ(s) and L(s, χ) 8.1 THE ZETA FUNCTION Definition The Riemann zeta function, which for σ > 1 is defined by the absolutely convergent series ζ(s) = n s Corollary For σ > 1, we have Also, we have n s = ζ(s) = p µ(n)n s = 1 ζ(s) = p λ(n)n s = ζ(2s) ζ(s) = p log ζ(s) = (1 p s ) 1, Λ(n) log n n s ζ (s) ζ(s) = Λ(n)n s ζ (s) = (log n)n s (1 p s ), (1 + p s ) 1 Theorem Letting B n denote the nth Bernoulli number, we have ζ( n) = ( 1) n B n+1 for n 0. n

40 CHAPTER 8. THE FUNCTIONS ζ(s) AND L(S, χ) In particular, ζ( 2n) = 0 for all n 1. Definition The function Γ(s) initially defined for σ > 0 has an analytic continuation to a meromorphic function whose only singularities are simple poles at the negative integers s = 0, 1,.... The residue of Γ at s = n is ( 1) n /n!. The gamma function is defined by Γ(s) = 0 e t t s 1 dt. Theorem (The Functional Equation) For each s 1, we have ζ(s) = 2 s π s 1 sin( 1 πs)γ(1 s)ζ(1 s) 2 Remark The functional equation takes the more symmetric form with the notation Φ(s) = Φ(1 s) for s 0, 1 Φ(s) = π s/2 Γ(s/2)ζ(s). Theorem The function ζ(s) is analytically continuable to a meromorphic function on the entire complex plane having as sole singularity a simple pole at s = 1, with residue 1. Theorem The function ζ(s) has no zero in the half-plane σ 1. In the half plane σ 0, it has simple zeros at the points 2n(n = 1, 2,...). Definition The zeros at the negative even integers are called the trivial zeros. The critical stip is where 0 < σ < 1. The non-trivial zeros are those that lie in the critical strip. The functional equation and the fact that ζ(s) is real for real s imply that the zeros are distributed symmetrically with respect to the line σ = 1 and the real 2 axis τ = 0. The function (s 1)ζ(s) is entire, but it is more convenient to consider instead ξ(s) = s(s 1)π s/2 Γ(s/2)ζ(s) = s(s 1)Φ(s) which in addition, satisifes the simpler functional equation ξ(s) = ξ(1 s) ξ(s) is an enire function, and not also that ξ(0) = ξ(1) = 1. ξ(s) 0 for σ 1 and for σ 0. In the critical strip, ξ(s) and ζ(s) have the same zeros. The general notation for a zero of ξ (i.e. a non-trivial zero of ζ) is ρ = β +iγ, and we set N(T ) = 1, ρ:0 γ T where all zeros ρ are counted with multiplicity. 34

41 CHAPTER 8. THE FUNCTIONS ζ(s) AND L(S, χ) Remark The Riemann hypothesis conjectures that all non-trivial zeros of ζ(s) lie on the line σ = 1/2. The generalised Riemann hypothesis conjectures that all the zeros of L-functions are situated on the critical line σ = 1/ APPROXIMATIONS AND BOUNDS Corollary The inequalities 1 σ 1 < ζ(σ) < σ σ 1 holds for all σ > 0. In particular, ζ(σ) < 0 for 0 < σ < 1. Corollary Let δ > 0 be fixed. Then ζ(s) = 1 s 1 + O(1) uniformly for s in the rectangle δ σ 2, t 1, and ( ) 1 ζ(s) (1 + τ 1 σ )min σ 1, log τ uniformly for δ σ 2, t 1. Corollary We have ζ( iτ) τ 1/6 log τ as τ Theorem Let a be such that 0 < a < 1. We have ζ(s) 3 τ 1 a for σ a, τ 1. 2a(1 a) In particular, for each positive constant c, we have for τ 2, σ 1 c/ log τ. ζ(s) log τ Corollary For any positive constant c and any integer k 0, we have for τ 2, σ 1 c/ log τ. Theorem For T 2, we have ζ (k) (s) (log τ ) k+1 N(T + 1) N(T ) log T 35

42 CHAPTER 8. THE FUNCTIONS ζ(s) AND L(S, χ) Theorem A T tends to infinity, we have N(T ) = T 2π log T 2π T 2π + O(log T ). Theorem There is an absolute constant c > 0 such that ζ(s) 0 for σ 1 c/ log τ. This is the classical zero-free region. Theorem Let c be the constant in previous theorem. If σ > 1 c/(2 log τ) and t 7/8, then ζ (s) log τ, ζ and log ζ(s) log log τ + O(1), 1 ζ(s) log τ. On the other hand, if 1 c/(2 log τ) < σ 2 and t 7/8, then ζ /ζ(s) = 1/(s 1) + O(1), log(ζ(s)(s 1)) 1, and 1/ζ(s) s DIRICHLET L-FUNCTIONS Definition For each character χ, with modulos q, the Dirichlet L-series is defined by L(s, χ) = χ(n)n s for σ > 1, with Euler expansion L(s, χ) = p (1 χ(p)p s ) 1 for σ > 1. In particular L(s, χ) 0 when σ > 1. For χ = χ 0, the above formula can be written as L(s, χ 0 ) = n s = (1 p s )ζ(s).,(n,q)=1 p q If χ χ 0, then χ(n) = 0 1 n kq for k = 1, 2,.... Hence χ(n) q. n x 36

43 CHAPTER 8. THE FUNCTIONS ζ(s) AND L(S, χ) Theorem If χ χ 0, then L(s, χ) is analytic for σ > 0. On the other hand, the function L(s, χ 0 ) is analytic in this half plane except for a simple pole at s = 1 with residue φ(q)/q. In either case, for σ > 1, and log L(s, χ) = L n=2 Λ(n) log n χ(n)n s L (s, χ) = Λ(n)χ(n)n s. Theorem (Dirichlet) If χ is a character modulo q with χ χ 0, then L(1, χ) 0. Corollary (Dirichlet s theorem) If (a, q) = 1, then there are infinitely many primes p a mod q, and indeed p a(q) log p p =. 37

44 CHAPTER 8. THE FUNCTIONS ζ(s) AND L(S, χ) 38

45 Chapter 9 The Prime Number Theorem 9.1 THEOREMS OF CHEBYSHEV AND MERTENS Theorem (Chebyshev) For x 2, we have x log 2 + O(log x) ψ(x) x log 4 + O ( (log x) 2) Corollary As x tends to infinity, we have x x (log 2 + o(1)) π(x) (log 4 + o(1)) log x log x. Theorem (Bertrand s Postulate) For every integer n > 3 there is at least one prime number p such that n < p < 2n 2. Theorem (Merten s first theorem) We have that log p = log x + O(1) p p x Theorem (Merten s second theorem) We have that (1 1/p) 1 = e γ log x + O(1) p x 9.2 THE PRIME NUMBER THEOREM Theorem The prime number theorem says that π(x) x log x Definition (Chebyshev s summatory functions) ψ(x) = n x Λ(n) = log lcm{n : n x}, 39

46 CHAPTER 9. THE PRIME NUMBER THEOREM θ(x) = p x log p Theorem For x 1 we have the representation ψ(x) = θ(x 1/k ) Definition We denote by π(x) the number of primes not exceeding x. Theorem For x 2, we have k=1 ψ(x) = θ(x) + O( x), π(x) = θ(x) ( ) x log x + O. (log x) 2 Corollary Let a, b, be constants such that 0 < a < log 2, b > log 4. For all sufficiently large x, we have ax θ(x) ψ(x) bx. Definition The logarithmic integral is defined to be li(x) = x 2 1 log u du Theorem There is a constant c > 0 such that ( ψ(x) = x + O ( θ(x) = x + O x exp(c log x) x exp(c log x) ), ), and uniformly for x 2. ( ) x π(x) = li(x) + O exp(c log x) 40