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arXiv · 2101.00279

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

Marco Aymone·Caio Bueno·Kevin Medeiros
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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.

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Marco Aymone, Caio Bueno, Kevin Medeiros. 2021-01-01. Multiplicative functions supported on the $k$-free integers with small partial sums. https://arxiv.org/abs/2101.00279

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