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→Explicit formula for the Riemann zeta function: math notation edits |
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Weil's form of the explicit formula states
:<math>
\begin{align} & {} \quad \Phi(1)+\Phi(0)-\sum_\rho\Phi(\rho) \\ & = \sum_{p,m} \frac{\log(p)}{p^{m/2}} (F(\log(p^m) + F(-\log(p^m)) - \frac{1}{2\pi} \int_{-\infty}^
\end{align}
▲\sum_{p,m}\frac{\log(p)}{p^{m/2}} (F(\log(p^m)+F(-\log(p^m)) -\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi(t)\Psi(t)dt</math>
</math>
where
*ρ runs over the non-trivial zeros of the zeta function
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*''m'' runs over positive integers
*''F'' is a smooth function all of whose derivatives are rapidly decreasing
*''φ'' is a Fourier transform of ''F'':
:: <math>\ *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 arise 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
==Generalizations==
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