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The full product of Dirichlet L-functions has no zero on Re s = 1

Statement

Let

$F_{q} (s) : = \prod_{χ m o d q} L (s, χ) .$

Then $F_{q}$ has no zero on the line $Re s = 1$ . Moreover, $F_{q}$ is meromorphic on a neighbourhood of the closed half-plane $Re s \geq 1$ , and any singularity at $s = 1$ is at most a simple pole.

Facts & Assumptions

Given: A modulus $q \geq 1$ and the product $F_{q} (s) = \prod_{χ m o d q} L (s, χ)$ .

[L1]

For unit classes, the character sum $\sum_{χ} χ (u)$ is $φ (q)$ when $u = 1$ and $0$ otherwise (Orthogonality relations for Dirichlet characters modulo q).

[L2]

Each $L (s, χ)$ has its Euler product on $Re s > 1$ (Euler product for Dirichlet L-functions).

[L3]

The principal factor has one simple pole at $1$ (The principal Dirichlet L-function factors through zeta).

[L4]

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

[L5]

An Euler product whose logarithmic Dirichlet coefficients are nonnegative cannot vanish on $Re s = 1$ if it has at most a simple pole at $1$ (Positive logarithmic Dirichlet series force boundary nonvanishing).

Proof

technique · direct

1.1

For $Re s > 1$ , [L2] gives $\log F_{q} (s) = \sum_{χ m o d q} \sum_{p, m \geq 1} χ (p)^{m} / (m p^{m s}) = \sum_{p, m \geq 1} (m p^{m s})^{- 1} \sum_{χ m o d q} χ (p^{m})$ . If $p ∣ q$ , then every term is $0$ . If $p ∤ q$ , then [L1] applied to the unit class of $p^{m}$ shows that the inner character sum is $φ (q)$ when $p^{m} \equiv 1 (m o d q)$ and $0$ otherwise. Hence the logarithmic coefficients of $F_{q}$ are nonnegative.

L1L2givenalgebra

2.1

By [L3] and [L4], the product $F_{q}$ is meromorphic on a neighbourhood of $Re s \geq 1$ , with at most a simple pole at $1$ and no other singularities on the boundary line. Step 1.1 therefore places $F_{q}$ under [L5], so $F_{q}$ has no zero on $Re s = 1$ . This proves both the nonvanishing claim and the stated meromorphic control at $s = 1$ .

step 1.1L3L4L5algebra∎

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Sources

Kiran S. Kedlaya, Notes on Analytic Number Theory, Theorem 3.7 (standard reference, not scraped)
Leonard Tomczak, Analytic Number Theory, Chapter 4 (standard reference, not scraped)