2020/09/29 by Coppola, Giovanni, Ghidelli, Luca
#11A25 #11N37 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2009.14121
We call RG(a):=∑q=1∞G(q)cq(a) the 'Ramanujan series', of coefficient G:N→C, where cq(a) is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define G a 'Ramanujan coefficient' when RG(a) converges pointwise, in all natural a. Then, RG:N→C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed F:N→C is :=G:N→ C|RG w-d, F=RG. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients G : their subset is called the 'multiplicative Ramanujan cloud', M. Our first main result, the "Finiteness convergence Theorem", for G multiplicative, among other properties equivalent to "RG well defined", reduces the convergence test to a finite set, i.e., RG w-d is equivalent to: RG(a) converges for all a dividing N(G)∈N, that we call the "Ramanujan conductor". Our second main result, the "Finite Euler product explicit formula", for multiplicative Ramanujan coefficients G, writes F=RG as a finite Euler product; thus, F is a semi-multiplicative function (following Rearick definition) and this product is the Selberg factorization for F. In particular, we have: F(a)=RG(a) converges absolutely, being finite (of length depending on non-zero p-adic valuations of a). Our third main result, called the "Multiplicative Ramanujan clouds", studies the important subsets of M; also giving, for all multiplicative F, the 'canonical Ramanujan coefficient' GF∈ M, proving: Any multiplicative F has a finite Ramanujan expansion with multiplicative coefficients.