L-functions and the Riemann Hypothesis (DRAFT) Keith Conrad Last updated: May 28, 2018 INTRODUCTION These are notes for a set of lectures on “L-functions and the Riemann Hy- pothesis” at the 2018 CTNT summer school. It describes basic properties of Dirichlet L-functions, with the Riemann zeta-function as an important special case. Properties we consider include special values, analytic continuation, and functional equation, and finally the Riemann Hypothesis for these L-functions: what it says, how it can be tested numerically, and some of its applications. For prerequisites, the reader is assumed to have already taken courses in complex analysis and abstract algebra. Some topics that should be familiar from complex analysis include: analytic and meromorphic functions, the idea of analytic continuation, and the residue theorem. From algebra the reader should know about homomorphisms and basic properties of rings. Keith Conrad May, 2018 CONTENTS 1 Introduction to the Zeta-function and Dirichlet L-functions1 1.1 The Riemann Zeta-function.....................1 1.2 Dirichlet L-functions.........................6 2 Some Topics from Analysis9 2.1 Rates of Growth...........................9 2.2 Infinite Series and Products..................... 13 2.3 Complex Analysis.......................... 18 2.4 Dirichlet Series............................ 36 3 Values at Positive Integers 45 3.1 Values of ζ(s) at Positive Even Integers.............. 45 3.2 Primitive Characters and Gauss Sums............... 49 3.3 Values of L(s; χ) at Positive Even or Odd Integers........ 61 4 Analytic Continuation and Functional Equation 71 4.1 Analytic Continuation of ζ(s) .................... 71 4.2 More Special Values of ζ(s) ..................... 76 4.3 Poisson Summation and the Theta-Function............ 78 v CHAPTER 1 INTRODUCTION TO THE ZETA-FUNCTION AND DIRICHLET L-FUNCTIONS 1.1 The Riemann Zeta-function The series X 1 1 1 1 = 1 + + + + :::; ns 2s 3s 4s n>1 converges absolutely when Re(s) > 1 since X 1 1 = < 1 ns nRe(s) n>1 R 1 Re(s) s on account of the convergence of 1 dx=x when Re(s) > 1. (Recall n = es log n, so jnsj = nRe(s).) Its behavior as a function of a complex variable was first studied by Riemann (1859), so the series is named after him and uses his notation. 2 The Riemann Zeta-function Definition 1.1. The Riemann zeta-function is X 1 ζ(s) := ns n>1 when Re(s) > 1. There are a host of similar functions, also called zeta-functions, introduced by mathematicians such as Dedekind, Epstein, Hasse, Hurwitz, Selberg, and Weil. Unless indicated otherwise, we will often write “the zeta-function” instead of “the Riemann zeta-function.” On the half-plane Re(s) > 1, the series ζ(s) converges uniformly on compact subsets, and this implies ζ(s) is holomorphic on this region. In addition to the series representation for ζ(s), there is a product represen- tation involving a product over prime numbers: for Re(s) > 1, Y 1 ζ(s) = : 1 − 1=ps p This product can be seen by expanding each factor 1=(1−1=ps) into a geometric series and multiplying out all the series (the manipulations can be justified when Re(s) > 1). Because the product for ζ(s) was first found by Euler, it is called the Euler product for ζ(s). The Euler product suggests there should be relations between ζ(s) and prime numbers, as indeed there are. The reason Riemann wrote his paper on the zeta-function was to sketch out a program for proving the Prime Number Theorem, which concerns the prime-counting function π(x) = #fp 6 x : p primeg: Theorem 1.2 (Prime Number Theorem). As x ! 1 we have x π(x) ∼ ; (1.1) log x which means the two sides have ratio tending to 1. This theorem was conjectured by Gauss and Legendre independently in the early 19th century and was first proved by Hadamard and de la Vallée Poussin independently in 1896 using the zeta-function and the ideas about it proposed by Riemann. Eventually it was shown that the Prime Number Theorem is equivalent to the nonvanishing of ζ(s) on the line Re(s) = 1. Another version Introduction to the Zeta-function and Dirichlet L-functions 3 of the Prime Number Theorem is Z x dt π(x) ∼ : 2 log t This is equivalent to (1.1) because the integral grows in the same way as x= log x when x ! 1 (take the limit of their ratio using L’Hospital’s rule). The integral on the right side is called the logarithmic integral of x and denoted Li(x). The series for ζ(s) diverges as s ! 1+, and Riemann discovered how to analytically continue ζ(s) around the point s = 1, by moving through the upper or lower half-planes into regions where the series that originally defines ζ(s) in Definition 1.1 no longer converges. Specifically, multiply ζ(s) by a power of π and a value of the Γ-function to define s Z(s) := π−s=2Γ ζ(s): 2 (See Section 2.3 for basic properties of the Gamma function Γ(s), including the location of its poles.) Although initially defined only for s with real part > 1, by using some ideas from Fourier analysis (the Poisson summation formula), Z(s) can be rewritten in the form Z 1 dx 1 Z(s) = !(x)(xs=2 + x(1−s)=2) − ; (1.2) 1 x s(1 − s) 2 where !(x) is the rapidly converging exponential series P e−πn x and this n>1 integral converges (and is holomorphic) at every s 2 C. So the right side of (1.2) extends Z(s) to a holomorphic function on all of C, except for simple poles 1 at s = 0 and s = 1 from the term . The equation s(1 − s) πs=2 Z(s) ζ(s) = Γ(s=2) now extends the zeta-function to all of C, except for a simple pole at s = 1. (The simple poles of Z(s) and Γ(s=2) at s = 0 cancel, so ζ(s) is holomorphic and nonvanishing at s = 0.) Because of the symmetry of (1.2) in s and 1 − s, we have Z(1 − s) = Z(s); which is called the functional equation. We will derive the analytic continuation 4 The Riemann Zeta-function and functional equation for the zeta-function in Section 4.1.1 The functional equation can be rewritten as πs ζ(1 − s) = 2(2π)−sΓ(s) cos ζ(s); (1.3) 2 a formula that expresses ζ(1 − s) in terms of ζ(s), but here the symmetry in s and 1 − s is broken. As we’ll see later, it is easy to show that ζ(s) has simple zeros at the negative even integers and all its other zeros lie in fs : 0 6 Re(s) 6 1g, which is called the critical strip. A more precise claim about the location of these remaining zeros is the famous Riemann Hypothesis (RH): The zeros of ζ(s) with real part between 0 and 1 all lie on the vertical line Re(s) = 1=2. The vertical line Re(s) = 1=2 is special for the zeta-function, being the line of symmetry for the transposition s 7! 1 − s in the functional equation. It is called the critical line. Error estimates on the difference jπ(x) − Li(x)j are closely related to the Riemann Hypothesis: RH is equivalent to the bound p 2 jπ(x) − Li(x)j 6 C x(log x) : p for a constant C > 0. The exponent in the factor x = x1=2 comes from the “1=2” defining the critical line. Note: It is false that jπ(x) − x= log xj 6 p C x(log x)2, so although x= log x and Li(x) grow in the same way, there are not always interchangeable. Exercises for Section 1.1 1. Write the letter ζ until you can do it easily and correctly (at least 25 times). 1To remove the two poles of Z(s) while retaining a functional equation, s(s − 1)π−s=2Γ(s=2)ζ(s) is a common substitute for Z(s), and many authors in fact designate this product as Z(s). For largely historical reasons, an additional factor of 1=2 is sometimes in- cluded as well. Introduction to the Zeta-function and Dirichlet L-functions 5 Z x dt x 2. Show that ∼ by verifying that 2 log t log x Z x dt log t lim 2 = 1: x!1 x= log x 3. a) Let pn denote the nth prime number, so p1 = 2 and p4 = 7. Use the Prime Number Theorem to show p n ∼ n log(pn) as n ! 1. b) Deduce from the Prime Number Theorem that log(π(x)) ∼ log x as x ! 1. Then show log(pn) ∼ log n, so by part a) we get an asymptotic formula for the nth prime: pn ∼ n log n: c) Show that the asymptotic relation pn ∼ n log n implies the Prime Num- ber Theorem. (Hint: For x > 2, let n = π(x), so pn 6 x < pn+1.) d) Let c > 0 be a positive number. Use part b) to show p lim [cn] = c: n!1 pn Here [y] denotes the greatest integer 6 y. This formula shows that the ratios of the primes are dense in the positive reals, in a somewhat con- structive manner. e) The convergence of the limit in part d) is very slow. To see this nu- merically, use a computer algebra package to compute p p =p until you p [ 2n] n find some n for which this ratio approximates 2 = 1:414 ::: correctly to two decimal places.