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

摘要

AbstractWe continue our study of convolution sums of two arithmetical functionsfandg, of the form∑n≤Nf(n)g(n+h), in the context of heuristic asymptotic formul?. Here, the integerh≥0is called, as usual, theshiftof the convolution sum. We deepen the study of finite Ramanujan expansions of generalf,gfor the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum offandg, namely in terms of its shifthand we compare this ‘‘shift Ramanujan expansion’’, with our previous finite expansions in terms of thefandgarguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formul?.]]>