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Math Help - Dirichlet Series Example

  1. #1
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    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?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    For example

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    There is no complex variable in your series.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by EinStone View Post
    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.
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  5. #5
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    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.
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