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

2021/01/01 by Marco Aymone, Aymone, Marco, Caio Bueno +3
Computer Science · Mathematics · #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.00279

openalex publication_date 2021/01/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We provide examples of multiplicative functions f supported on the k-free integers such that at primes f(p)=± 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+(1)/(2))+ε}), for all ε>0. This generalizes to the k-free integers the results of Aymone, `` A note on multiplicative functions resembling the Möbius function'', J. Number Theory, 212 (2020), pp. 113--121.

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