12 1. Introduction Example 1.3.1 (The Riemann zeta function). The Riemann zeta function, usually denoted by ζ(s), is perhaps the most famous L- function: ζ(s) = n=1 1 ns = 1 + 1 2s + 1 3s + · · · . The reader may already know some values of ζ. For example ζ(2) = 1 n2 is convergent by the p-series test, and its value is π2/6 (this value can be computed using Fourier analysis and Parseval’s equality). The connection between ζ(s) and number theory comes from the fact that ζ(s) has an Euler product: ζ(s) = n=1 1 ns = p prime 1 1 p−s = 1 1 2−s · 1 1 3−s · 1 1 5−s · · · . This Euler product is not difficult to establish (Exercise 1.4.8) and has the very interesting consequence that any information on the distribution of the zeros of ζ(s) can be translated into information about the distribution of prime numbers among the natural numbers. Example 1.3.2 (Dirichlet L-function). Let a, N N be relatively prime integers. Are there infinitely many primes p of the form a+kN (i.e., p a mod N ) for k 0? The answer is yes and this fact, known as Dirichlet’s theorem on primes in arithmetic progressions, was first proved by Dirichlet using a particular kind of L-function that we know today as a Dirichlet L-function. Let N 0. A Dirichlet character (modulo N ) is a function χ : (Z/NZ)× that is a homomorphism of groups, i.e., χ(nm) = χ(n)χ(m) for all n, m (Z/NZ)×. Notice that χ(n) C and χ(n)ϕ(N) = 1 for all gcd(n, N) = 1. Therefore, χ(n) must be a root of unity. We extend χ to Z as follows. Let a Z. If gcd(a, N) = 1, then χ(a) = χ(a mod N). Otherwise, if gcd(a, N) = 1, then χ(a) = 0. A Dirichlet L-function is a function of the form L(s, χ) = n=1 χ(n) ns ,
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