Explicit Formula - Weil's Explicit Formula

Weil's Explicit Formula

There are several slightly different ways to state the explicit formula. Weil's form of the explicit formula states

\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}^\infty\varphi(t)\Psi(t)\,dt \end{align}

where

  • ρ runs over the non-trivial zeros of the zeta function
  • p runs over positive primes
  • m runs over positive integers
  • F is a smooth function all of whose derivatives are rapidly decreasing
  • φ is a Fourier transform of F:
  • 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
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 * counted with multiplicities, so the poles at 0 and 1 are counted as zeros of order −1.

Read more about this topic:  Explicit Formula

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