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## Weil positivity

The explicit formula of prime number theory relates the zeros of a zeta function (the spectral side) with the counting function of divisors (the geometric side). It was first discovered by Riemann in 1859 (and probably already earlier), and was then established rigorously by de la Vallée-Poussin in 1896 [dV1, dV2].1

The explicit formula was first viewed as a solution to the problem of counting prime numbers. At the time, the spectral side was considered relatively well understood, since it was easily established that the Riemann zeta function does not vanish on the line Re s = 1.

Nowadays, the opposite opinion may be more widely held: the spectral side is not very well understood, and it may be possible to use the explicit formula ...

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