A round of Pintz to celebrate oscillations in sums

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.