Multiplicative functions supported on the $k$-free integers with small partial sums

TL;DR

This paper constructs examples of multiplicative functions supported on k-free integers with small partial sums, improving bounds under the Generalized Riemann Hypothesis, extending previous results to a broader class of functions.

Contribution

It introduces new examples of multiplicative functions supported on k-free integers with refined bounds on their partial sums, especially under GRH, generalizing prior work.

Findings

Existence of multiplicative functions with partial sums o(x^{1/k})

Under GRH, improved bounds to o(x^{1/(k+1/2)+ε})

Extension of previous results to broader classes of functions

Abstract

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+\epsilon})$, for all $\epsilon>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}\"{o}bius function'', J. Number Theory, 212 (2020), pp. 113--121.