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Residues in the von Mangoldt contour shift

Statement

Assume countable choice.

For $x > 1$ , shifting $- ζ^{'} (s) x^{s} / (ζ (s) s)$ left crosses residues $x (s = 1), - \frac{x^{ρ}}{ρ} (s = ρ), - \frac{1}{2} \log (1 - x^{- 2}) (s = - 2, - 4, \dots), - \frac{ζ^{'} (0)}{ζ (0)} (s = 0) .$ Zeros are counted with multiplicity and $x^{ρ} = \exp (ρ \log x)$ uses real $\log x$ .

Facts & Assumptions

[L1]

The zeros of zeta in $Re s \leq 0$ occur exactly at the negative even integers; in particular, $s = 0$ is not a zero (The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip).

[L2]

Zeta satisfies $ζ (s) = 2^{s} π^{s - 1} \sin (π s / 2) Γ (1 - s) ζ (1 - s)$ as an identity of meromorphic functions (The Riemann zeta function satisfies the classical sine-gamma functional equation).

[L3]

Gamma has poles only at the nonpositive integers and has no zeros, while zeta has no zeros on $Re s \geq 1$ (Meromorphic continuation of Gamma, Gamma has no zeros, The Riemann zeta function has no zeros on the closed half-plane $Re s \geq 1$ , except for its pole at $1$ ).

Proof

Given: Countable choice, $x > 1$ and the meromorphic continuation of zeta.

1.1givenalgebra

A simple pole of zeta at $1$ makes $- ζ^{'} / ζ$ have residue $1$ ; a zero $ρ$ of multiplicity $m$ makes it have residue $- m$ . Multiplication by $x^{s} / s$ gives the first two entries.

2.1L1L2L3step 1.1algebra∎

By [L1], the remaining zeros crossed on the nonpositive real axis occur at $- 2 k$ . At $s = - 2 k$ , the sine in [L2] has a simple zero, while all its other factors are finite and nonzero by [L3]; hence these zeros are simple. Their residues sum to $\sum_{k \geq 1} x^{- 2 k} / (2 k) = - \frac{1}{2} \log (1 - x^{- 2})$ . Also by [L1], zeta is nonzero at $0$ , so the pole of $1 / s$ gives the final entry.

Depends on

Used by

The von Mangoldt residue table Example
A smoothed von Mangoldt explicit formula Theorem
The truncated von Mangoldt explicit formula Theorem

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Sources

Kiran S. Kedlaya, Analytic Number Theory, §10.1 (standard reference, not scraped)