The structure of multiplicative functions with small partial sums
Discrete analysis (2020)
Cet article a éte moissonné depuis la source Scholastica
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 $v0$, 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 $γ_1$, $\dots$, $γ_m$ such that $f(p)\approx -p^{iγ_1}-\cdots-p^{-iγ_m}$ on average. The numbers $γ_j$ 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.
@article{DAS_2020_a15,
author = {Dimitris Koukoulopoulos and K. Soundararajan},
title = {The structure of multiplicative functions with small partial sums},
journal = {Discrete analysis},
year = {2020},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DAS_2020_a15/}
}
Dimitris Koukoulopoulos; K. Soundararajan. The structure of multiplicative functions with small partial sums. Discrete analysis (2020). http://geodesic.mathdoc.fr/item/DAS_2020_a15/