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Explicit formulae for L-function

  1. Aug 24, 2008 #1
    let be the L-function [tex] F(s)= \sum_{n=1}^{\infty} X(n) n^{-s} [/tex] with a single pole at s=1

    then my question is if one can define [tex] \frac{-F'(s)}{F(s)}= \sum_{n=1}^{\infty} a(n) n^{-s} [/tex],

    then taking an inverse Mellin transform we get

    [tex] \sum_{n \le x} a(n) = x - \sum_{r} r^{-1} x^{r} [/tex]

    the question is , what are the a(n) ,

    if X(n)=1 for every n then F(s) is just Riemann zeta and a(n) /\(n) Mangoldt function,

    the question is , can the result be generalized for every X(n)
  2. jcsd
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