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# Weil's positivity criterion

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Andr\'{e} Weil [
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MR 14,727e
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MR 10,104g
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specifically illustrates the dependence between primes and zeros.
Let $h$ be an
even function which is holomorphic in the strip $|\Im t|\le
1/2+\delta$ and satisfying $h(t)=O((1+|t|)^{-2-\delta})$ for some
$\delta>0$, and let
$$g(u)=\frac{1}{2\pi}\int_{-\infty}^\infty h(r)e^{-i u r}~dr.$$
Then we have the following duality between primes and zeros:
$$
\sum_{\gamma}h(\gamma)=2h(\tfrac{i}2) -g(0) \log \pi
+\frac{1}{2\pi} \int_{-\infty}^\infty
h(r)\frac{\Gamma'}{\Gamma}(\tfrac14+\tfrac 12 i
r)~dr-2\sum_{n=1}^\infty \frac{\Lambda(n)}{\sqrt{n}}g(\log n).$$
In this formula, a zero is written as $\rho=1/2+i\gamma$ where
$\gamma\in \mathbb C$; of course RH is the assertion that all of
the $\gamma$ are real. Using this duality Weil gave a \begin{rawhtml}criterion\end{rawhtml} for RH. \begin{rawhtml}

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