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 (J. Number Theory, 212:113-121, 2020).
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Notes
We recall that \(\zeta \) stands for the classical Riemann zeta function and for a Dirichlet character \(\chi \), \(L(s,\chi )\) stands for the classical L-function \(L(s,\chi )=\sum _{n=1}^\infty \frac{\chi (n)}{n^s}\). The Riemann Hypothesis for \(\zeta \) states that all zeros \(\zeta (s_0)=0\) with real part \(Re(s_0)>0\) have real part equals 1/2. The Generalized Riemann Hypothesis is the same statement for \(L(s,\chi )\).
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We would like to thank the anonymous referee for a careful reading of the paper and for useful suggestions that improved the exposition.
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Marco Aymone is supported by CNPq - Grant Universal Number 403037/2021-2; Caio Bueno was financed in part by the scholarship from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance code 001; Kevin Medeiros was financed by the Fapemig scholarship.
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Aymone, M., Bueno, C. & Medeiros, K. Multiplicative functions supported on the k-free integers with small partial sums. Ramanujan J 59, 713–728 (2022). https://doi.org/10.1007/s11139-022-00618-9
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DOI: https://doi.org/10.1007/s11139-022-00618-9