I would like to know if the Dirichlet inverse $L(s,g)$ of a series $L(s,f)$ ($f(1)\neq 0$) with finite abscissa of convergence also has a finite abscissa of convergence? Is there a specific criteria to ensure it is the case?
Asked
Active
Viewed 196 times
1 Answers
1
No.
For instance, suppose $f(1) = 1$ and $f(2) = -1$, while $f(n) = 0$ for all $n \geq 3$. Then $F(s) = \sum f(n) n^{-s} = 1 - 2^{-s}$, which is absolutely convergent everywhere.
But it's Dirichlet inverse is $G(s) = \frac{1}{1 - 2^{-s}}$, which only converges for $\Re s > 0$.
davidlowryduda
- 91,687
-
1your example does not work, because we want f(n) to have a finite abscissa of convergence. – usere5225321 Oct 10 '16 at 13:11
-
@usere5225321 Then think of $G(s)$ as the original and $F(s)$ as the inverse. – davidlowryduda Oct 10 '16 at 13:27