On multiplicative functions with small partial sums

TL;DR

This paper investigates multiplicative functions with small partial sums, showing they approximate sums involving zeros of their Dirichlet series, extending prior results and offering insights into their prime value behavior.

Contribution

It proves that divisor-bounded multiplicative functions with small partial sums approximate sums over zeros of their Dirichlet series, extending previous work by Koukoulopoulos and Soundararajan.

Findings

Multiplicative functions with small partial sums are closely related to zeros of their Dirichlet series.

The paper extends existing results to a broader class of functions beyond those bounded by 1.

Provides a new perspective on the prime values of multiplicative functions in relation to their zeros.

Abstract

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the Landau-Selberg-Delange method. In this paper, we are interested in the opposite direction. In particular, we prove that when $f$ is a suitable divisor-bounded multiplicative function with small partial sums, then $f(p)\approx-p^{i\gamma_1}-\ldots-p^{i\gamma_m}$ on average, where the $\gamma_j$'s are the imaginary parts of the zeros of the Dirichet series of $f$ on the line $\Re(s)=1$. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where $|f|\leqslant 1$.