# Dirichlet Series Example

• Apr 28th 2011, 07:21 AM
EinStone
Dirichlet Series Example
Hey, Im looking for an example of a Dirichlet series $\sum_{n=1}^{\infty}\frac{a_n}{n^s},s \in \mathbb{C}$, whose convergence abscissa differs from its absolute convergence abcissa, but the difference is strictly less than 1.

Can you help me, and is it clear what Im looking for?
• Apr 28th 2011, 09:24 AM
FernandoRevilla
• Apr 28th 2011, 09:26 AM
EinStone
There is no complex variable in your series.
• Apr 28th 2011, 09:34 AM
FernandoRevilla
Quote:

Originally Posted by EinStone
There is no complex variable in your series.

My example satisfies all the given hypothesis. If you meant s complex but not real then you have incorrectly expressed the question.
• Apr 28th 2011, 09:46 AM
EinStone
I meant an example with an arbitrary complex s, for instance $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^s}$ converges for Re s > 0 and converges absolutely for Re s > 1. SO the difference of the abscissa is exactly 1. I am now looking for an example where the difference is strictly between 0 and 1.