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

TL;DR

This paper investigates the convergence properties of Ramanujan series, introduces the concept of Ramanujan clouds, and establishes finite Euler product formulas for multiplicative coefficients, advancing the understanding of Ramanujan expansions.

Contribution

It provides a finite convergence criterion, explicit Euler product formulas, and the existence of finite multiplicative Ramanujan expansions for functions.

Findings

Convergence reduces to checking finitely many divisors.

Finite Euler product formulas for multiplicative functions.

Existence of a canonical multiplicative Ramanujan coefficient.

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.