For a function $f: \mathbb N \to \mathbb C$ , let $\sigma_c(f) , \sigma_a(f)$ denote the abscissa of convergence and the abscissa of absolute convergence respectively of the Dirichlet series $\sum_{n=1}^\infty \dfrac {f(n)}{n^s}$ .
Can we characterize all arithmetic functions $f:\mathbb N \to \mathbb C$ with $f(1)\ne 0$ , such that $\sigma_a(f) \ge \sigma_c(f^{-1})$ , where $f^{-1}$ denotes the "inverse" of $f$ with respect to Dirichlet convolution ( the inverse exists as we have assumed $f(1)\ne 0$ ) ? If this is difficult , then how about a characterization of arithmetic functions $f$ satisfying $\sigma_a(f) \ge \sigma_a(f^{-1})$ ? If these cannot be done very well, then can we at least characterize such functions among the multiplicative functions ?
Definitely, any completely multiplicative arithmetic function satisfies the condition, as for such a function $f$ , $f^{-1}(n)=\mu(n)f(n)$ ; and since $|\mu(n)f(n)| \le |f(n)|$ , we get $\sigma_a(f) \ge \sigma_a(f^{-1})\ge \sigma_c(f^{-1})$ . But I don't know what other kind of functions satisfy the condition I have mentioned .
Please help . Thanks in advance