A round of Pintz to celebrate oscillations in sums

Abstract

Abstract We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -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 $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the Möbius function, but we also outline the utility of this method in general.