On Multiplicative Functions with Small Partial Sums

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