Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf 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 ??

via Recent Questions - Mathematics - Stack Exchange http://math.stackexchange.com/questions/294436/theta-series-and-riemann-hypothesis