1. 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?

2. For example

$\displaystyle\sum_{n=1}^{+\infty}\dfrac{(-1)^{n+1}+ 0i}{n^2}$

3. There is no complex variable in your series.

4. 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.

5. 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.