Explicit formulae for L-functions: Difference between revisions

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where <math>F,G</math> are the Fourier transforms of <math>f,g</math>.

At a first look, it seems to be a formula for functions only, but in fact in many cases it also works when <math>g</math> is a distribution. Hence, by setting <math display="block">g(u) = \sum_{n=1}^\infty \Lambda(n) \left[ \delta(u+\ln n) + \delta(u-\ln n) \right] , </math> where <math>\delta(u)</math> is the [[Dirac delta function|Dirac delta]], and carefully choosing a function <math>f</math> and its Fourier transform, we get the formula above.

==Explicit formulae for other arithmetical functions==

{{unreferenced section|date=September 2020}}

The Riemann-Weil formula<ref>{{Cite web |title=the Riemann-Weil explicit formula |url=https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/weilexplicitformula.htm |access-date=2023-06-14 |website=empslocal.ex.ac.uk}}</ref> can be generalized to arithmetical functions other than the [[von Mangoldt function]]. For example for the Möbius function we have

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

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)} + \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.

In all cases the sum is related to the imaginary part of the Riemann zeros <math display="inline"> \rho = \frac{1}{2}+i \gamma </math> and the function ''h'' is related to the test function ''g'' by a Fourier transform, <math display="inline"> g(u) = \frac{1}{2\pi} \int_{-\infty}^\infty h(x) \exp(-iux) </math>.

For the divisor function of zeroth order <math> \sum_{n=1}^\infty \sigma_0 (n) f(n) = \sum_ {m=-\infty}^\infty \sum_{n=1}^\infty f(mn) </math>.{{clarify|reason=The relation of this to the preceding material needs to be explained.|date=September 2020}}

Using a test function of the form <math>g(x) = f(ye^{x}) e^{ax} </math> for some positive ''a'' turns the Poisson summation formula into a formula involving the Mellin transform. Here ''y'' is a real parameter.

==Generalizations==

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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==