Multiplicative Ramanujan coefficients of null-function

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.