On multiplicative functions which are small on average

Dimitris Koukoulopoulos

doi:10.1007/s00039-013-0235-6

On multiplicative functions which are small on average

Number Theory 2017-06-12 v5

Authors: Dimitris Koukoulopoulos

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Abstract

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n<x} f(n) \ll x/(\log x)^A$ , $x>2$ , for some $A>2$ , then either $f(p)$ is small on average or $f$ pretends to be $\mu(n)n^{it}$ for some $t$ .

Keywords

multiplicative functions divisor function

Cite

@article{arxiv.1111.2659,
  title  = {On multiplicative functions which are small on average},
  author = {Dimitris Koukoulopoulos},
  journal= {arXiv preprint arXiv:1111.2659},
  year   = {2017}
}

Comments

51 pages. Slightly strengthened Theorem 1.2 and simplified its statement. Removed Remark 1.3. Other minor changes and corrections. To appear in Geom. Funct. Anal

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