# Number theoretical theta function

1. Jun 6, 2008

### mhill

my question is given the theta function

$$F(x)= \sum_{n=1}^{\infty}\mu (n) e^{-\pi n^{2}x}$$ (1)

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

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

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