Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors.
Does such formula really exist? If yes, was it proven?
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It is a sum, exact but not efficient for finding the value. See http://store.doverpublications.com/0486417409.html – Will Jagy Jan 24 '15 at 00:02
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2see http://en.wikipedia.org/wiki/Prime-counting_function – Will Jagy Jan 24 '15 at 00:07
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One might claim that the real point is the precise relation between primes and the zeros of zeta, as made clearer in Guinand's and Weil's extensions of Riemann's "Explicit formula", to be a sort of Fourier duality between the two. I find the relation more interesting than either questions about counting primes, or any sort of "raw" version of questions about the zeros... – paul garrett Jan 24 '15 at 00:26