Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

in the paper section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series

$$ F(\phi(x))= x^{1/2}\sum_{n=0}^{\infty}\phi (nx) $$ x >0

of course if we take the Mellin transform of this we will get the formula

$$ G(1/2+is)\zeta(1/2+is) $$

$$ G(s)= \int_{0}^{\infty}dtF(\phi(t))t^{s-1} $$

then the author affirms that the function vanishes at the Riemann zeros only

i know all the properties of this function but why does the authoer introduce it ?? he wants to give an spectral interpretation but how does the spectral interpretation appears from this series ??

share|cite|improve this question


$\zeta(\sigma+2i\pi f)$ is the Fourier transform of $\sum_{n=1}^\infty \delta(t - \ln n) n^{-\sigma}$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.