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

Famous quotes containing the words weil, explicit and/or formula:

    “Every perfect life is a parable invented by God.”
    —Simone Weil (1909–1943)

    “Like dreaming, reading performs the prodigious task of carrying us off to other worlds. But reading is not dreaming because books, unlike dreams, are subject to our will: they envelop us in alternative realities only because we give them explicit permission to do so. Books are the dreams we would most like to have, and, like dreams, they have the power to change consciousness, turning sadness to laughter and anxious introspection to the relaxed contemplation of some other time and place.”
    —Victor Null, South African educator, psychologist. Lost in a Book: The Psychology of Reading for Pleasure, introduction, Yale University Press (1988)

    “Ideals possess the strange quality that if they were completely realized they would turn into nonsense. One could easily follow a commandment such as “Thou shalt not kill” to the point of dying of starvation; and I might establish the formula that for the proper functioning of the mesh of our ideals, as in the case of a strainer, the holes are just as important as the mesh.”
    —Robert Musil (1880–1942)