On multiplicative functions which are small on average

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 $t$.

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

References in corpus (1)

• Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

Cited by in corpus (3)

• Partial sums of biased random multiplicative functions
• On multiplicative functions with small partial sums
• A pretentious proof of Linnik's estimate for primes in arithmetic progressions