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In [[mathematics]], the '''[[Closed-form expression|explicit formulae]] for [[L-function|L-functions]]''' are relations between sums over the complex number zeroes of an [[L-function]] and sums over [[Prime number|prime powers]], introduced by {{harvtxt|Riemann|1859}} for the [[Riemann zeta function]]. Such explicit formulae have been applied also to questions on bounding the [[discriminant of an algebraic number field]], and the [[conductor of a number field]].
==Riemann's explicit formula==
In his 1859 paper "[[On the Number of Primes Less Than a Given Magnitude]]" Riemann sketched an explicit formula (it was not fully proven until 1895 by [[Hans Carl Friedrich von Mangoldt|von Mangoldt]], see below) for the normalized prime-counting function {{math|π<sub>0</sub>(''x'')}} which is related to the [[prime-counting function]] {{math|π(''x'')}} by<ref>{{Cite web |title=Explicit Formulae (L-function) |url=https://encyclopedia.pub/entry/32287 |access-date=2023-06-14 |website=encyclopedia.pub |language=en}}</ref>
:<math>\pi_0(x) = \frac{1}{2} \lim_{h\to 0} \left[\,\pi(x+h) + \pi(x-h)\,\right]\,,</math>
which takes the [[arithmetic mean]] of the limit from the left and the limit from the right at discontinuities.{{efn|The original prime counting function can easily be recovered via <math>~\pi(x) = \pi_0(x+1)~</math> for all <math>~x \ge 3~.</math>}} His formula was given in terms of the related function
:<math>f(x) = \pi_0(x) + \frac{1}{2}\,\pi_0(x^{1/2}) + \frac{1}{3}\,\pi_0(x^{1/3}) + \cdots</math>
in which a prime power {{math|''p''<sup>''n''</sup>}} counts as {{frac|1|{{mvar|n}}}} of a prime. The normalized [[prime-counting function]] can be recovered from this function by
:<ref>{{Cite journal |last=Li |first=Xian-Jin |date=2004-04 |title=Explicit formulas for Dirichlet and Hecke $L$-functions |url=https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-48/issue-2/Explicit-formulas-for-Dirichlet-and-Hecke-L-functions/10.1215/ijm/1258138394.full |journal=Illinois Journal of Mathematics |volume=48 |issue=2 |pages=491–503 |doi=10.1215/ijm/1258138394 |issn=0019-2082}}</ref><math>\pi_0(x) = \sum_n\frac{1}{n}\,\mu(n)\,f(x^{1/n}) = f(x) - \frac{1}{2}\,f(x^{1/2}) - \frac{1}{3}\,f(x^{1/3}) - \frac{1}{5}\,f(x^{1/5}) + \frac{1}{6}\,f(x^{1/6}) - \cdots,</math>
where {{math|''μ''(''n'')}} is the [[Möbius function]]. Riemann's formula is then
:<math>f(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}(x^\rho) - \log(2) + \int_x^\infty \frac{dt}{~t\,(t^2-1)~\log(t)~}</math>
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==Weil's explicit formula ==
There are several slightly different ways to state the explicit 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>. [[André Weil]]'s form of the explicit formula states
:<math>
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The terms in the formula arise in the following way.
*The terms on the right hand side come from the [[logarithmic derivative]] of <math display="block">\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 product|Euler factor]] at infinity).
*The left-hand side is a sum over all zeros of ''ζ''<sup> *</sup> counted with multiplicities, so the poles at 0 and 1 are counted as zeros of order −1.
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where {{math|Λ}} is the [[von Mangoldt function]].
So that the [[Fourier transform]] of the non trivial zeros is equal to the primes power symmetrized plus a minor term. Of course, the sum involved are not convergent, but the trick is to use the unitary property of Fourier transform which is that it preserves scalar product:
: <math>\int_{-\infty}^\infty f(u) g^*(u) \, du = \int_{-\infty}^\infty F(t) G^*(t) \, dt</math>
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{{unreferenced section|date=September 2020}}
The Riemann-Weil formula<ref>{{
: <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>
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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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