On Multiplicative Functions with Small Partial Sums

Abstract

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 $\textrm {Re}(s)=1$. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where $|f|\leqslant 1$.