2025/09/29 by Coppola, Giovanni
#11N37 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2509.24456
In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function F, building upon a good decay of its coefficients G; this, gains log-powers w.r.t. the trivial bound for G and precisely log1+η, where the present parameter η>0 is real. This property alone has important consequences for all the F having a G.R.E. : mainly, 1) the Eratosthenes Transform F' of our F is infinitesimal (see in Theorem 1); 2) when η>1 (an enhanced decay) we have uniqueness of G (actually, these are the classic Wintner-Carmichael coefficients, see Th.2); 3) we get a bound for F (in Th.3); 4) an important new class of arithmetic functions F can't have a G.R.E. (see Th.4). These are a generalization of Correlations; which in this way, if are, say, a kind of "far from constants", may not have a G.R.E., whence, a fortiori, can't have the R.E.E.F. This is the Ramanujan Exact Explicit Formula, that we introduced with Prof. Ram Murty.