Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, II

摘要

We continue our study of convolution sums of two arithmetical functions $f$and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristicasymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\itshift} of the convolution sum. We deepen the study of finite Ramanujanexpansions of general $f,g$ for the purpose of studying their convolution sum.Also, we introduce another kind of Ramanujan expansion for the convolution sumof $f$ and $g$, namely in terms of its shift $h$ and we compare this \lq \lqshifted Ramanujan expansion\rq \rq, with our previous finite expansions interms of the $f$ and $g$ arguments. Last but not least, we give examples ofsuch shift expansions, in classical literature, for the heuristic formul\ae.