Explicit formulae for L-functions: Difference between revisions

Revision as of 13:13, 14 June 2023

Revision as of 15:15, 14 June 2023

Line 1:

In [[mathematics]], the '''[[Closed-form expression|explicit formulae]] for [[L-function|L-functions]]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==

Line 7:

:<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 2004-04 |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|''&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>

Line 28:

==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>