A round of Pintz to celebrate oscillations in sums

TL;DR

This paper revisits Pintz's method linking sums of arithmetic functions to zero-free regions of L-functions, explicitly demonstrating how arithmetic data can inform us about L-function zeros, exemplified with the Riemann zeta-function.

Contribution

It explicitly formulates Pintz's general result, showing how arithmetical information can be used to deduce properties of L-function zeros, reversing traditional approaches.

Findings

Explicit connection between sums of arithmetic functions and zero-free regions.

Application to the Riemann zeta-function and Möbius function sums.

Outline of the method's utility in broader contexts.

Abstract

We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of $L$-functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the M\"obius function, but we also outline the utility of this method in general.