my question is given the theta function(adsbygoogle = window.adsbygoogle || []).push({});

[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

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# Number theoretical theta function

Can you offer guidance or do you also need help?

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