the "encoding" of the distribution of primes by the zeros of the Riemann zeta function

" Riemann showed that the [nontrivial] zeros of [the zeta function] determine the fluctuations in the density of primes (that is their importance) and conjectured that they all have real part 1/2."

M.V. Berry, from "Quantum Chaology", The Bakerian Lecture 1987, Proceedings of the Royal Society of London A 413 (1987) 183–198

"...the complex zeros of the Riemann zeta-function induce the local variations in the distribution of the primes."
Hans Riesel from Prime Numbers and Computer Methods for Factorisation (Birkhaüser, 1994)

There are two approaches:

the more common approach              the more elegant approach

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