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Line 38: ==Applications== Riemann's original use of the explicit formula way to give an exact formula for the number of primes less than a given number. To do this, take ''F''(log(''y'')) to be ''y''<sup>1/2</sup>/log(''y'') for 0 ≤ ''y'' ≤ ''x'' and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than ''x''. The main term on the left if ''Φ''(1); which turns out to be the dominant terms of the [[prime number theorem]], and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function ''F'' does not satisfy the smoothness condition.) 0≤''y''≤''x'' and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than ''x''. The main term on the left if Φ(1); which turns out to be the dominant terms of the [[prime number theorem]], and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function ''F'' does not satisfy the smoothness condition.) ==Hilbert-Pólya conjecture==

Explicit formulae for L-functions: Difference between revisions