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A nonreal Dirichlet L-function is nonzero at one

Statement

If $χ$ is a nonreal Dirichlet character, then $L (1, χ) \neq 0$ .

Facts & Assumptions

Given: A nonreal Dirichlet character $χ$ modulo $q$ .

[L1]

The full product $\prod_{ψ m o d q} L (s, ψ)$ has no zero on the line $Re s = 1$ (The full product of Dirichlet L-functions has no zero on Re s = 1).

[L2]

The principal Dirichlet $L$ -function has a simple pole at $s = 1$ (The principal Dirichlet L-function factors through zeta).

[L3]

Every nonprincipal Dirichlet $L$ -function is holomorphic on $Re s > 0$ (Nonprincipal Dirichlet L-functions are holomorphic on Re s greater than 0).

[A1]

Complex conjugation sends a Dirichlet character to another Dirichlet character $\overset{ˉ}{χ}$ , and $L (1, \overset{ˉ}{χ}) = \overline{L (1, χ)}$ .

Proof

technique · contradiction

1.1

Suppose $L (1, χ) = 0$ . Then [A1] gives $L (1, \overset{ˉ}{χ}) = 0$ as well. Because $χ$ is nonreal, the characters $χ$ and $\overset{ˉ}{χ}$ are distinct, so these are two different vanishing factors in the full finite product at $s = 1$ .

A1givenassume-contra

2.1

In the full product over all characters, [L2] contributes order $- 1$ at $s = 1$ , while step 1.1 contributes at least $+ 1$ from each of the distinct factors $L (s, χ)$ and $L (s, \overset{ˉ}{χ})$ . Every remaining nonprincipal factor is holomorphic at $1$ by [L3], so it contributes order at least $0$ . Hence the total product has order at least $+ 1$ at $1$ , meaning a zero there. This contradicts [L1]. Therefore $L (1, χ) \neq 0$ .

L1L2L3step 1.1discharge-contradiction∎

Depends on

Used by

Nonprincipal Dirichlet L-functions are nonzero at one Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kiran S. Kedlaya, Notes on Analytic Number Theory, Theorem 3.10 (standard reference, not scraped)
Leonard Tomczak, Analytic Number Theory, Chapter 4 (standard reference, not scraped)