The structure of multiplicative functions with small partial sums

TL;DR

This paper characterizes the structure of multiplicative functions with small partial sums, linking their behavior to zeros of their Dirichlet series and generalizing previous results to broader classes of functions.

Contribution

It proves a converse theorem showing that small partial sums imply the function's values approximate sums involving zeros of its Dirichlet series, extending prior work.

Findings

Functions with small partial sums are approximated by sums over zeros of their Dirichlet series.

The structure of such functions is determined by finitely many real parameters related to these zeros.

Generalizes previous results to functions bounded by divisor functions, not just |f|1|.

Abstract

The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,\dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely many real numbers $\gamma_1$, $\dots$, $\gamma_m$ such that $f(p)\approx -p^{i\gamma_1}-\cdots-p^{-i\gamma_m}$ on average. The numbers $\gamma_j$ correspond to ordinates of zeroes of the Dirichlet series associated to $f$, counted with multiplicity. This generalizes a result of the first author, who handled the case when $|f|\le 1$ in previous work.