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→Explicit formula for the Riemann zeta function: explanation |
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==Explicit formula for the Riemann zeta function==
There are several slightly different ways to state the explicit formula.
:<math>\Phi(1)+\Phi(0)-\sum_\rho\Phi(\rho)
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*Φ(1/2 + it) = φ(t)
*Ψ(t) = -log(π) + Re(ψ(1/4 + it/2)), where ψ is the [[digamma function]] Γ′/Γ
Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function
is the set of prime powers plus some elementary factors.
The terms in the formula aries in the following way.
*The terms on the right hand side come from the logarithmic derivative of
:<math>\zeta^*(s)= \Gamma(s/2)\pi^{-s/2}\prod_{p}\frac{1}{1-p^{-s}}</math>
: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.
==Hilbert-Pólya conjecture==
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