Polya's integral criterion

Polya (see Collected Works, Volume 2, Paper 102, section 7) gave a number of integral criteria for Fourier transforms to have only real zeros. One of these, applied to the Riemann $\xi$-function, is as follows.

The Riemann Hypothesis is true if and only if

\begin{displaymath}\int_{-\infty}^\infty\int_{-\infty}^\infty \Phi(\alpha)\Phi(\...
...a)x }
e^{(\alpha-\beta)y}(\alpha-\beta)^2~d\alpha ~d\beta \ge 0\end{displaymath}

for all real $x$ and $y$ where

\begin{displaymath}\Phi(u)=2\sum_{n=1}^\infty (2 n^4\pi^2e^{\frac 9 2 u}-3 n^2 \pi e^{\frac 5 2 u})e^{-n^2 \pi e^{2 u}}\end{displaymath}

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