Geodesic
Convolution of periodic multiplicative functions and the divisor problem
Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 1966-1984

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DOI

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.
DOI : 10.4153/S0008414X2400066X
Mots-clés : Dirichlet divisor problem, correlations in the divisor problem, omega bounds, periodic multiplicative functions 
Aymone, Marco; Maiti, Gopal; Ramaré, Olivier; Srivastav, Priyamvad. Convolution of periodic multiplicative functions and the divisor problem. Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 1966-1984. doi: 10.4153/S0008414X2400066X
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     title = {Convolution of periodic multiplicative functions and the divisor problem},
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     pages = {1966--1984},
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