André Weil [
MR 14,727e] proved the following explicit formula (see also A. P. Guinand [
MR 10,104g] which
specifically illustrates the dependence between primes and zeros.
Let be an
even function which is holomorphic in the strip
and satisfying
for some
, and let

Then we have the following duality between primes and zeros:

In this formula, a zero is written as where ; of course RH is the assertion that all of the are real. Using this duality Weil gave a criterion for RH.

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for The Riemann Hypothesis.