Logarithmic mean values of multiplicative functions

Abstract: A general mean-value theorem for multiplicative functions taking values in the unit disc was given by Wirsing (1967) and Halász (1968). We consider a multiplicative function f belonging to a certain class of arithmetical functions and let F(s) be the associated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f. More precisely, we give estimates in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence, we obtain a non-trivial zero-free region for partial sums of L-functions belonging to our class. We also report on some recent work showing that this zero free region is optimal. This is joint work with Arindam Roy (UNC Charlotte).

number theory

Audience: researchers in the topic

Number theory during lockdown

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