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<h1 align=center> Polya's integral criterion
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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 
$$\int_{-\infty}^\infty\int_{-\infty}^\infty \Phi(\alpha)\Phi(\beta) e^{i(\alpha+\beta)x }
e^{(\alpha-\beta)y}(\alpha-\beta)^2~d\alpha ~d\beta \ge 0$$
for all real $x$ and $y$ where
$$\Phi(u)=2\sum_{n=1}^\infty (2 n^4\pi^2e^{\frac 9 2 u}-3 n^2 \pi e^{\frac 5 2 u})e^{-n^2 \pi e^{2 u}}$$\begin{rawhtml}
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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