On Ramanujan expansions with multipliative coefficients, II

In an earlier paper \cite{[L]} we showed that if $\widehat g$ is a multiplicative arithmetic function such that $|\widehat g(p)|<1$ for any prime number $p$, then the convergence of the series $\sum_{n} \widehat g(n)\mu(n)$ yields the convergence of the subseries $\sum_{\gcd(n,a)=1} \widehat g(n)\mu(n)$ for any positive integer $a$. In this way, by exploiting well known relationships between the M\"obius function $\mu$ and the Ramanujan sum $c_n(a)$, we gained also the convergence of the Ramanujan series $\sum_n\widehat g(n)c_n(a)$. Here we give a new result in this direction when $|\widehat g(p)|\ge 1$ for some prime $p$. Further, in \cite{[L]} we also proved that if $\widehat g$ is multiplicative and such that $\sum_{\gcd(n,p)=1}\widehat g(n)\mu(n)$ converges for some prime number $p$, then the Ramanujan series $\sum_n\widehat g(n)c_n(p^r)$ converges for any integer $r\ge 0$. In this paper we prove a partial reverse of this implication when $\widehat g(p^{r+1})\not=\widehat g(p^{r})$ for some positive integer $r$.