Explicit formulae for L-functions: Difference between revisions

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{{Short description|Mathematical concept}}

In [[mathematics]], the '''[[Closed-form expression|explicit formulae]] for [[L-functionsfunction]]s''' 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|π₀(''x'')}} which is related to the [[prime-counting function]] {{math|π(''x'')}} by{{cn|date=February 2024}}

:<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''^''n''}} 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=April 2004 |title=Explicit formulas for Dirichlet and Hecke $L$-functions |journal=Illinois Journal of Mathematics |volume=48 |issue=2 |pages=491–503 |doi=10.1215/ijm/1258138394 |issn=0019-2082|doi-access=free }}</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|''&mu;''(''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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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>\quad\sigma > 1\,, \quad \psi(x) = \sum_{p^k \le x} \log p\,,

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

and the RHS is obtained from the [[residue theorem]], and then converting it into the formula that Riemann himself actually sketched.

This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part:<ref name=Ing77>Ingham (1990) p.77</ref>

:<math>\sum_\rho\frac{x^\rho}{\rho} = \lim_{T \rightarrowto \infty} S(x,T) \quad</math> {{pad|2em}} where {{pad|2em}} <math>\quad S(x,T) = \sum_{\rho:\left|\Im \rho\right| \le T} \frac{x^\rho}{\rho}\,.</math>

The error involved in truncating the sum to {{math|''S''(''x'',''T'')}} is always smaller than {{math|ln(''x'')}} in absolute value, and when divided by the [[natural logarithm]] of {{mvar|x}}, has absolute value smaller than {{math|{{frac|''x''|''T''}}}} divided by the distance from {{mvar|x}} to the nearest prime power.<ref>[https://math.stackexchange.com/q/497949 Confused about the explicit formula for ψ0(x)]</ref>

==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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*<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.

*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 &Psi; coming from the gamma factor (the [[Euler product|Euler factor]] at infinity).

*The left-hand side is a sum over all zeros of ''ζ''^&nbsp;* counted with multiplicities, so the poles at 0 and 1 are counted as zeros of order &minus;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-WeylWeil formula<ref>{{clarifyCite web |reasontitle=Athe formulaRiemann-Weil byexplicit this nameformula is not mentioned in the article|url=https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/weilexplicitformula.htm |access-date=September2023-06-14 2020|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)} + \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.

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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:<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==

*[[Selberg trace formula]]

*[[Selberg zeta function]]

==Footnotes==

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*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Sur les "formules explicites" de la théorie des nombres premiers | trans-title=On "explicit formulas" in the theory of prime numbers | mr=0053152 | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=Tome Supplémentaire | pages=252–265 | zbl=0049.03205 | language=fr }}

*{{Citation | last1 = von Mangoldt | first1 = Hans | title=Zu Riemanns Abhandlung "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" | journal = [[Journal für die reine und angewandte Mathematik]] | volume=114 | year=1895 | pages=255–305 | jfm=26.0215.03 | language=de | issn=0075-4102 | mr=1580379 | trans-title=On Riemann's paper "The number of prime numbers less than a given magnitude" }}

*{{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"

* Garcia J.J Mellin Convolution and its Extensions, Perron Formula and Explicit Formulae doi=10.20944/preprints201801.0020.v1

* https://encyclopediaofmath.org/wiki/M%C3%B6bius_function#:~:text=The%20M%C3%B6bius%20function%20is%20an,M%C3%B6bius%20in%201832

==Further reading==