Multiplicative Ramanujan coefficients of null-function

TL;DR

This paper classifies multiplicative Ramanujan coefficients of the null-function, revealing new exotic types and their convergence properties, extending Ramanujan's and Hardy's classical expansions.

Contribution

It provides a complete classification of multiplicative Ramanujan coefficients of the null-function, introducing exotic and weakly exotic types and analyzing their convergence behaviors.

Findings

Exotic Ramanujan coefficients are characterized by convergence hypotheses.

Only exotic coefficients have absolute convergence.

Many examples of Ramanujan coefficients are provided.

Abstract

The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/\varphi(q))c_q(a)$, given by Hardy ($\varphi:=$ Euler's totient function). Both converge pointwise (not absolutely) in N. A $G:$N $\rightarrow $C is called a Ramanujan coefficient, abbrev. R.c., iff (if and only if) $\sum_{q=1}^{\infty}G(q)c_q(a)$ converges in all $a\in $N; given $F:$N $\rightarrow $C, we call $<F>$, the set of its R.c.s, the Ramanujan cloud of $F$. Our Main Theorem in arxiv:1910.14640, for Ramanujan expansions and finite Euler products, implies a complete Classification for multiplicative Ramanujan coefficients of $0$. Ramanujan's $G_R(q):=1/q$ is a normal arithmetic function $G$, i.e., multiplicative with $G(p)\neq 1$ on all primes $p$; while Hardy's $G_H(q):=1/\varphi(q)$ is a sporadic $G$, namely multiplicative, $G(p)=1$ for a finite set of $p$, but there's no $p$ with $G(p^K)=1$ on all integers $K\ge 0$ (Hardy's has $G_H(p)=1$ iff $p=2$). The $G:$N $\rightarrow $C multiplicative, such that there's at least a prime $p$ with $G(p^K)=1$, on all $K\ge 0$, are defined to be exotic. This definition completes the cases for multiplicative $0-$Ramanujan coefficients. The exotic ones are a kind of new phenomenon in the $0-$cloud (i.e., $<0>$): exotic Ramanujan coefficients represent $0$ only with a convergence hypothesis. The not exotic, apart from the convergence hypothesis, require in addition $\sum_{q=1}^{\infty}G(q)\mu(q)=0$ for normal $G\in <0>$, while sporadic $G\in <0>$ need $\sum_{(q,P(G))=1}G(q)\mu(q)=0$, $P(G):=$product of all $p$ making $G(p)=1$. We give many examples of R.c.s $G\in <0>$; we also prove that the only $G\in <0>$ with absolute convergence are the exotic ones; actually, these generalize to the weakly exotic, not necessarily multiplicative.