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Nonprincipal Dirichlet L-functions are holomorphic on Re s greater than 0

Statement

If $χ \neq χ_{0}$ is a Dirichlet character, then the Dirichlet series $L (s, χ) = \sum_{n \geq 1} χ (n) n^{- s}$ converges for every $Re s > 0$ and defines a holomorphic function there.

Facts & Assumptions

Given: A nonprincipal Dirichlet character $χ$ .

[L1]

The partial sums $A (x) = \sum_{1 \leq n \leq x} χ (n)$ satisfy $A (x) = O (1)$ (Nonprincipal Dirichlet character partial sums are bounded).

[L2]

If $A (x) = O (x^{θ})$ , then $\sum_{n \geq 1} a_{n} n^{- s} = s \int_{1}^{\infty} A (x) x^{- s - 1} d x$ for $Re s > θ$ (Dirichlet series from arithmetic functions admit the Abel-summation integral formula).

[L3]

The Dirichlet $L$ -function is the Dirichlet series $\sum_{n \geq 1} χ (n) n^{- s}$ (Dirichlet L-functions).

Proof

technique · direct

1.1

Apply [L2] to the coefficients $a_{n} = χ (n)$ with $θ = 0$ . By [L1], the summatory function is bounded, so for every $Re s > 0$ one has $L (s, χ) = s \int_{1}^{\infty} A (x) x^{- s - 1} d x$ , and the integral converges absolutely and locally uniformly on each half-plane $Re s \geq ε > 0$ because $A (x) = O (1)$ and $x^{- s - 1} = O (x^{- ε - 1})$ .

L1L2L3givenalgebra

2.1

A locally uniformly convergent parameter integral of holomorphic integrands is holomorphic in the parameter. Hence the right-hand side of step 1.1 defines a holomorphic function on $Re s > 0$ , and on the smaller half-plane $Re s > 1$ it agrees with the defining Dirichlet series [L3]. Therefore $L (s, χ)$ is holomorphic on $Re s > 0$ .

step 1.1L3algebra∎

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8 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 3.5 (standard reference, not scraped)
Andrew V. Sutherland, Number Theory I, Proposition 18.20 (standard reference, not scraped)