What are some major implications of a proof of the Riemann hypothesis?
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"Tons" of major applications can be found at this MO-question. – Dietrich Burde Nov 28 '16 at 19:08
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1st of all : the Riemann explicit formula $\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho}-\ln 2\pi$ relates directly the zeros of $\zeta(s)$ to the regularity of the distribution of prime numbers. – reuns Dec 02 '16 at 16:08