a Dirichlet L-function
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A Dirichlet L-function is a close cousin of the zeta function, built to study primes not among all integers but within an arithmetic progression — primes of the form 4k+1, or 7k+3, and so on. The ingredient that selects a progression is a Dirichlet character chi modulo q: a function on the integers that is periodic with period q, completely multiplicative (chi(mn) = chi(m) chi(n)), and zero on integers sharing a factor with q. Feeding it into a Dirichlet series gives L(s, chi) = sum over n >= 1 of chi(n) / n^s. When chi is the trivial (all-ones, modulo 1) character you recover zeta itself, so L-functions are a family with zeta as the simplest member.
Two zeta-like features carry over and do the heavy lifting. First, because chi is multiplicative, L(s, chi) has an Euler product over primes, product over p of 1/(1 - chi(p) p^(-s)), so it too encodes prime information factor by factor. Second, each L(s, chi) analytically continues to the whole plane and satisfies its own functional equation relating s and 1 - s (through a gamma factor and a quantity called the Gauss sum). For a non-principal character the continuation is even entire — no pole at all — which is a crucial technical advantage over zeta.
The reason these were invented is one of the jewels of the subject: Dirichlet's theorem on primes in arithmetic progressions. As long as a and q share no common factor, the progression a, a+q, a+2q, ... contains infinitely many primes. The proof hinges on showing that L(1, chi) is non-zero for every non-principal character chi modulo q — exactly the kind of non-vanishing statement that, for zeta on the line Re s = 1, gives the prime number theorem. The generalised Riemann hypothesis is the conjecture that every Dirichlet L-function, like zeta, has all its non-trivial zeros on the line Re s = 1/2. A caveat: proving L(1, chi) is non-zero for COMPLEX characters is easy, but for real (quadratic) characters it is genuinely delicate — that single non-vanishing is the heart of Dirichlet's argument.
The simplest non-trivial example is the character modulo 4: chi(n) = 1 if n = 1 mod 4, chi(n) = -1 if n = 3 mod 4, and chi(n) = 0 if n is even. Then L(s, chi) = 1 - 1/3^s + 1/5^s - 1/7^s + ... , and at s = 1 this is the Leibniz series 1 - 1/3 + 1/5 - 1/7 + ... = pi/4. Since L(1, chi) = pi/4 is non-zero, Dirichlet's theorem gives infinitely many primes of each form 4k+1 and 4k+3.
An L-function is zeta with a character inserted; its non-vanishing at s = 1 powers Dirichlet's theorem on primes in progressions.
For the principal character, L(s, chi) is essentially zeta and inherits its pole at s = 1; for a non-principal character L(s, chi) is entire (no pole). The whole strength of Dirichlet's theorem rests on the harder fact L(1, chi) is non-zero, which is subtle precisely for real characters.