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

We continue our study of convolution sums of two arithmetical functions f and g, of the form ∑n≤Nf(n)g(n+h), in the context of heuristic asymptotic formulæ. Here, the integer h≥0 is called, as usual, the shift of the convolution sum. We deepen the study of finite Ramanujan expansions of general f,g for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of f and g, namely in terms of its shift h and we compare this ''shift Ramanujan expansion'', with our previous finite expansions in terms of the f and g arguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formulæ.