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→Explicit formula for the Riemann zeta function: explanation |
→Explicit formula for the Riemann zeta function: applications |
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:with the terms corresponding to the prime ''p'' coming from the Euler factor of ''p'', and the term at the end involving Ψ coming from the gamma factor (the Euler factor at infinity).
*The left hand side is a sum over all zeros of ζ<sup>*</sup> counted with multiplicities, so thepoles at 0 and 1 are counted as zeros of order −1.
==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''. 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==
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