On multiplicative functions with small partial sums

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 $f(p)\approx -p^{i{\gamma }_1}-\dots -p^{i{\gamma }_m}$ on average, where the ${\gamma }_j$’s are the imaginary parts of the zeros of the Dirichet series of $f$ on the line $\mathrm{\Re }(s)=1$. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where $\mathrm{\mid }f\mathrm{\mid }⩽1$.