Convergence of Ramanujan expansions, I [Multiplicativity on Ramanujan clouds]

Abstract

We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ 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 $R_G(a)$ converges pointwise, in all natural $a$. Then, $R_G:$N$\to$C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed $F:$N$\to$C is $<F>:=${$G:N\to C|R_G \; w-d, F=R_G$}. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients $G$ : their $<F>$ subset is called the 'multiplicative Ramanujan cloud', $<F>_M$. Our first main result, the "Finiteness convergence Theorem", for $G$ multiplicative, among other properties equivalent to "$R_G$ well defined", reduces the convergence test to a finite set, i.e., $R_G$ w-d is equivalent to: $R_G(a)$ converges for all $a$ dividing $N(G)\in$N, that we call the "Ramanujan conductor". Our second main result, the "Finite Euler product explicit formula", for multiplicative Ramanujan coefficients $G$, writes $F=R_G$ 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: $F(a)=R_G(a)$ 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 $<F>_M$; also giving, for all multiplicative $F$, the 'canonical Ramanujan coefficient' $G_F\in <F>_M$, proving: Any multiplicative $F$ has a finite Ramanujan expansion with multiplicative coefficients.