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Integral equation for Riesz function

  1. Feb 24, 2010 #1
    i was thinking about an integral equation for Riesz function

    [tex] e^{-x}-1= \int_{0}^{\infty}\frac{dt}{t}frac(\sqrt (xt)Riesz(1/t) [/tex]

    apparently if should work , in fact if i take Mellin transform to both sides it gives me

    [tex] \Gamma (s)= \frac{\Gamma(s+1)}{\zeta (-2s)} \frac{\zeta(-2s)}{s} [/tex]

    if i make the essay (by professor wolf) for the Riesz function [tex] Riesz(x)= x^{1/4}sin(A- \gamma /2 log(x)) [/tex] , with 'gamma' being the imaginary part of the First non-trivial zero for the Riemman zeta function i get the result

    0=1 ¡¡ , and i do not know how to follow.

    Euler's formula [tex]2i sin(log(x))=x^{i}+x^{-i} [/tex]

    and the fractional part representation (valid for 1>s>0 )

    [tex] -\zeta(s)=s\int_{0}^{\infty}dt frac(1/t)t^{s-1} [/tex]
     
  2. jcsd
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