On Multiplicative Functions with Bounded Partial Sums

TL;DR

This paper investigates whether certain multiplicative functions with restricted properties can have bounded partial sums, showing that weakening multiplicativity on a finite set of primes allows for such boundedness.

Contribution

It demonstrates that multiplicative functions with weakened conditions on a finite set of primes can have bounded partial sums, expanding understanding of their behavior.

Findings

Bounded partial sums are possible under weakened multiplicativity.

Modification of characters at finite places affects partial sum bounds.

Results connect multiplicative functions with finite set restrictions.

Abstract

Consider a multiplicative function f(n) taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes p in some finite set P, then the answer is yes. We also discuss a result of Bronstein that shows that functions modified from characters at a finite number of places.