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We study a certain class of arithmetic functions that appeared in Klurman'sclassification of $\pm 1$ multiplicative functions with bounded partial sums,c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodicand $1$-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 aconjecture by the first author. As a byproduct of our proof, we studied thecorrelation between $\Delta(x)$ and $\Delta(\theta x)$, where $\theta$ is afixed 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 itquantitative this decorrelation in terms of this measure.

Convolution of periodic multiplicative functions and the divisor problem