Complex Valued Multiplicative Functions with Bounded Partial Sums

Aymone, Marco

doi:10.1007/s00574-022-00306-8

Bull Braz Math Soc, New Series

2022

DOI: 10.1007/s00574-022-00306-8

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Complex Valued Multiplicative Functions with Bounded Partial Sums

Marco Aymone¹

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2024

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Convolution of periodic multiplicative functions and the divisor problem

Aymone,

Maiti,

Ramaré

et al. 2024

Can. J. Math.-J. Can. Math.

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We study a certain class of arithmetic functions that appeared in Klurman’s classification of $\pm 1$ multiplicative functions with bounded partial sums; c.f., Comp. Math. 153(2017), 2017, no. 8, 1622–1657. These functions are periodic and $1$ -pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum _{n\leq x}(f_1\ast f_2)(n)=\Omega (x^{1/4})$ . This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $\Delta (x)$ and $\Delta (\theta x)$ , where $\theta $ is a fixed real number. We prove that there is a nontrivial correlation when $\theta $ is rational, and a decorrelation when $\theta $ is irrational. Moreover, if $\theta $ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

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Convolution of periodic multiplicative functions and the divisor problem

Aymone,

Maiti,

Ramaré

et al. 2024

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

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Complex Valued Multiplicative Functions with Bounded Partial Sums

Cited by 1 publication

References 9 publications

Convolution of periodic multiplicative functions and the divisor problem

Convolution of periodic multiplicative functions and the divisor problem

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