Explicit formulae for L-functions: Difference between revisions

Revision as of 09:17, 13 August 2023

Revision as of 05:55, 14 August 2023

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:<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 |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|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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*{{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 }}

* 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