Summary: 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 \(\sum_{n\leq x}(f_1\ast f_2)(n)=\Omega(x^{1/4})\). 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 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.