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Number theoretical theta function

  1. Jun 6, 2008 #1
    my question is given the theta function

    [tex] F(x)= \sum_{n=1}^{\infty}\mu (n) e^{-\pi n^{2}x} [/tex] (1)

    it can be proved that this function F(x) is related to the Dirichlet series for Mobius function

    [tex] \sum_{n=1}^{\infty} \mu (n) n^{-s} = \frac{ \pi ^{s/2}}{\Gamma (s/2) } \int_{0}^{\infty}dx F(x) x^{s/2 -1 } [/tex]

    then my idea would be trying to obtain a functional equation for the function F(x) above in the form [tex] F(x)= x^{a} F(x^{b}) [/tex] to see if we can obtain a functional equation for the sum (1) to extend it to negative values
  2. jcsd
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