Convolution of periodic multiplicative functions and the divisor problem

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 (8), 2017, pp. 1622-1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then ${\sum }_{n\le x}(f_1\ast f_2)(n)=\mathrm{\Omega }(x^{1\mathrm{/}4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $\mathrm{\Delta }(x)$ and $\mathrm{\Delta }(\theta x)$, where $\theta$ is a fixed real number. We prove that there is a non-trivial 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.