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Conjecture about Dirichlet series.

  1. Feb 8, 2008 #1
    Hi, i hope it is not a crack theory

    i had the idea when reading Ramanujan resummation, i belive that for a Dirichlet series which converges for Re (s) > a , with a a positive real number then we can obtain a regularized sum of the divergent series in the form:

    [tex] \sum_{n >1}a_{n}n^{-s}- C(s-a)^{b} [/tex]

    with C, a and b real numbers , in case a_n are all 1 we recover the result

    [tex] \sum_{n >1}n^{-s}- (s-1)^{-1} [/tex]

    however i believe that 'Ramanujan resummation of series' given by

    [tex] \sum_{n >1}a_{n}n^{-s}-\int_{1}^{\infty}a(x)x^{-s} [/tex]

    must be satisfied no matter what kind of numerical series is.
  2. jcsd
  3. Feb 13, 2008 #2
    I think you're saying that the tail integral which is characteristic of Ramanujan resummation can be put in the form c(s-a)^b always for a, b, c constants. However can it be shown that the integral will evaluate in the given form always?
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