Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted