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Dirichlet series

  1. Feb 23, 2007 #1
    Let be the series in the form [tex] g(s)= \sum_{1 \le n } |\Lambda (n) |^{2} n^{-s} [/tex] where lambda is Von Mangoldt function, my question is how could i get an exact or at least almost exact expresion for g(s) . My other question is how could i obtainthe Mellin transform of the function [tex] \Lambda (n+2) \Lambda (n+1) [/tex] i have tried sum by parts but got no results.
     
    Last edited: Feb 23, 2007
  2. jcsd
  3. Feb 24, 2007 #2
    of course i tried (wrong ??) with the property, given a Dirchlet series [tex] g(s)= \sum_{1 \le n} f(n)n^{-s} [/tex] then [tex] -g'(s)/g(s)= \sum_{1 \le n} f(n) \Lambda (n) n^{-s} [/tex] but i think this does not work unless f(n) is multiplicative, however could it be used at least as a good approximation or modifying it a bit coul work??
     
    Last edited: Feb 24, 2007
  4. Feb 24, 2007 #3

    matt grime

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    Welcome back, Jose.
     
  5. Feb 24, 2007 #4
    readng Tom Apostol's 'Introduction to Analytic number theory' the desired series is just [tex] AT^{-1} \int_{-T}^{T}dt | \frac{ \zeta ' (a+it)}{\zeta (a+it)} |^{2} = \sum_{1 \le n} | \Lambda (n) |^{2} n^{2a} [/tex] , with a>1 and T tending to infinity. but i don't know the value of the integral, could someone provide an approximate value of the integral over t (-T,T) above ??, thank you.
     
  6. Feb 24, 2007 #5
    How many accounts has this guy made on here?
     
  7. Feb 24, 2007 #6
    I have also tried using partial summation so:

    [tex] \sum_{n \le x } \Lambda (n+2) \Lambda (n) = B(x)= \Lambda (x+2) \Psi (x) - \sum_{n \le x } \Psi (x) ( \Lambda (x+2) - \Lambda (x+1) ) [/tex]

    so [tex] g(s)= \sum_{1 \le n} \Lambda (n+2) \Lambda (n) n^{-s}= s \int_{1}^{\infty}B(x) x^{- (s+1)} [/tex]

    to obtain 'g(s)' any hint please?? thanx.
     
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