Explicit formulae for L-functions: Difference between revisions

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The first rigorous proof of the aforementioned formula was given by von Mangoldt in 1895: it started with a proof of the following formula for the [[Chebyshev's function]] {{mvar|ψ}}&nbsp;<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/ExplicitFormula.html Explicit Formula] on MathWorld.</ref>

:<math>\psi_0(x) = \dfrac{1}{2\pi i} \int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}\, ds = x - \sum_\rho\frac{~x^\rho\,}{\rho} - \log(2\pi) -\dfrac{1}{2}\log(1-x^{-2})</math>

where the LHS is an inverse [[Mellin transform]] with

:<math>\sigma > 1\,, \quad \psi(x) = \sum_{p^k \le x} \log p\,,

\quad \text{and} \quad \psi_0(x) = \frac{1}{2} \lim_{h\to 0} (\psi(x+h) + \psi(x-h))</math>

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*<math>\Psi(t) = - \log( \pi ) + \operatorname{Re}(\psi(1/4 + it/2))</math>, where <math>\psi</math> is the [[digamma function]] {{math|Γ<big>&prime;</big>/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. Once this is said, the formula comes from the fact that the Fourier transform is a [[unitary operator]], so that a scalar product in time ___domain is equal to the scalar product of the Fourier transforms in the frequency ___domain.

The terms in the formula arise in the following way.

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Also for the Liouville function we have

: <math> \sum_{n=1}^\infty \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\rho}\frac{h( \gamma)\zeta(2 \rho )}{\zeta'( \rho)} + \frac{1}{2\zeta (1/2)}\int_{-\infty}^\infty dx \, g(x) .</math>

For the Euler-Phi function the explicit formula reads

: <math> \sum_{n=1}^{\infty} \frac{\varphi (n)}{\sqrt{n}}g(\log n) = \frac{6}{\pi ^2} \int_{-\infty}^\infty dx \, g(x) e^{3x/2} + \sum_\rho \frac{h( \gamma)\zeta(\rho -1 )}{\zeta '( \rho)} + \frac{1}{2}\sum_{n=1}^\infty \frac{\zeta (-2n-1)}{\zeta'(-2n)} \int_{-\infty}^\infty dx \, g(x)e^{-x(2n+1/2)} .</math>

Assuming Riemann zeta function has only simple zeros.

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:<math> \sum_\rho F(\rho) = \operatorname{Tr}(F(\widehat T )).\!</math>

Development of the explicit formulae for a wide class of L-functions was given by {{harvtxt|Weil|1952}}, who first extended the idea to [[local zeta-function]]s, and formulated a version of a [[generalized Riemann hypothesis]] in this setting, as a positivity statement for a [[generalized function]] on a [[topological group]]. More recent work by [[Alain Connes]] has gone much further into the functional-analytic background, providing a trace formula the validity of which is equivalent to such a generalized Riemann hypothesis. A slightly different point of view was given by {{harvtxt|Meyer|2005}}, who derived the explicit formula of Weil via [[harmonic analysis]] on [[Adele ring|adelic]] spaces.

==See also==

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*{{Citation | last1 = Meyer | first1 = Ralf | title=On a representation of the idele class group related to primes and zeros of ''L''-functions | journal = [[Duke Math. J.]] | volume=127 | number=3 | year=2005 | pages=519–595 | zbl=1079.11044 | issn=0012-7094 | doi=10.1215/s0012-7094-04-12734-4 | mr=2132868 | arxiv=math/0311468 | s2cid = 119176169 }}

*{{citation | last = Zagier | first = Don |author-link= Don Zagier | doi = 10.1007/bf03351556 | issue = S2 | journal = [[The Mathematical Intelligencer]] | pages = 7–19 | title = The first 50 million prime numbers | volume = 1 | year = 1977| s2cid = 37866599 }}

* https://www.gsjournal.net/Science-Journals/Research%20Papers/View/9990 Moreta, Jose Javier Garcia:"On the evaluation of certain arithmetical functions of number theory and their sums and a generalization of riemann-weil formula"

==Further reading==