en>Bgwhite: WP:CHECKWIKI error fix. Broken bracket problem. Do general fixes and cleanup if needed. - using AWB (9507)

2013-10-07T20:17:47Z

WP:CHECKWIKI error fix. Broken bracket problem. Do general fixes and cleanup if needed. - using AWB (9507)

← Older revision		Revision as of 22:17, 7 October 2013
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	~~Hi there~~, ~~I am Alyson Pomerleau~~ and ~~I believe it seems fairly good when you say it~~. He is an ~~information officer~~. ~~Her family life in Alaska but her husband wants them to move~~. The ~~preferred hobby for him~~ and ~~his kids~~ is to ~~perform lacross~~ and ~~he would never~~ give ~~it up~~.<br><br>~~My web page~~ ... ~~real psychic ([http:/~~/~~www~~.~~rosextv~~.~~com/uprofile~~.~~php?UID~~=~~389761 http~~:~~//www.rosextv.com~~])		In [[mathematics]], the '''Chowla–Selberg formula''' is the evaluation of a certain product of values of the [[Gamma function]] at rational values in terms of values of the [[Dedekind eta function]] at imaginary quadratic irrational numbers. The result was essentially found by {{harvs\|txt\|author-link=Mathias Lerch\|last=Lerch\|year=1897}} and rediscovered by {{harvs\|txt\|last=Chowla\|author1-link=Sarvadaman Chowla\|Chowla\|author2-link=Atle Selberg\|last2=Selberg\|year1=1949\|year2=1967}}.

			==Statement==

			In logarithmic form, the Chowla–Selberg formula states that in certain cases the sum

			: <math> \frac{w}{4}\sum_r \chi(r)\log \Gamma\left( \frac{r}{D} \right) = \frac{h}{2}\log(4\pi\sqrt{\|D\|})
			+\sum_\tau\log\left(\sqrt{\Im(\tau)}\|\eta(\tau)\|^2\right)
			</math>

			can be evaluated using the [[Kronecker limit formula]]. Here χ is the [[quadratic residue symbol]] modulo ''D'', where ''−D'' is the [[discriminant]] of an imaginary [[quadratic field]]. The sum is taken over 0 < ''r'' < ''D'', with the usual convention χ(''r'') = 0 if ''r'' and ''D'' have a common factor. The function η is the [[Dedekind eta function]], and ''h'' is the class number, and ''w'' is the number of roots of unity.

			The origin of such formulae is now seen to be in the theory of [[complex multiplication]], and in particular in the theory of periods of an [[abelian variety of CM-type]]. This has led to much research and generalization. In particular there is an analog of the Chowla–Selberg formula for [[p-adic number]]s, involving a [[p-adic gamma function]], called the [[Gross–Koblitz formula]].

			The Chowla–Selberg formula gives a formula for a finite product of values of the eta functions. By combining this with the theory of [[complex multiplication]], one can give a formula for the individual absolute values of the eta function as
			:<math>\Im(\tau)\|\eta(\tau)\|^4 = \frac{\alpha}{4\pi\sqrt{\|D\|}} \prod_r\Gamma(r/\|D\|)^{\chi(r)\frac{w}{2h}}</math>
			for some algebraic number α.

			==Examples==

			Using the reflection formula for the gamma function gives:

			*<math>\eta(i) = 2^{-1}\pi^{-3/4}\Gamma(1/4)</math>

			==See also==
			* [[Multiplication theorem]]

			{{Reflist}}

			==References==

			*{{Citation \| last1=Chowla \| first1=S. \| last2=Selberg \| first2=Atle \| title=On Epstein's zeta function. I \| jstor=88112 \| mr=0030997 \| year=1949 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=35 \| pages=371–374}}
			*{{Citation
			\| title=On Epstein's Zeta-function
			\| first2=Atle
			\| last2=Selberg
			\| author2-link=Atle Selberg
			\| first=Sarvadaman
			\| last=Chowla
			\| journal=[[Journal für die reine und angewandte Mathematik]]
			\| year=1967
			\| volume=227
			\| pages=86–110
			\| doi=10.1515/crll.1967.227.86
			\| mr=0215797
			\| issue=227
			}}
			*{{Citation
			\| last=Lerch
			\| first=Mathias
			\| author-link=Mathias Lerch
			\| title=Sur quelques formules relatives au nombre des classes
			\| journal=[[Bulletin des Sciences Mathématiques]]
			\| volume=21
			\| year=1897
			\| pages=290–304
			}}
			*{{Citation
			\| last=Schappacher
			\| first=Norbert
			\| title=Periods of Hecke characters
			\| series=Lecture Notes in Mathematics
			\| year=1988
			\| publisher=[[Springer-Verlag]]
			\| location=Berlin
			\| volume=1301
			\| mr=0935127
			}}

			{{DEFAULTSORT:Chowla-Selberg formula}}
			[[Category:Number theory]]
			[[Category:Gamma and related functions]]


			{{numtheory-stub}}

en>SD5bot: BOT: replacing template(s) per discussion about misleading template name

2012-04-02T07:46:46Z

BOT: replacing template(s) per discussion about misleading template name

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Hi there, I am Alyson Pomerleau and I believe it seems fairly good when you say it. He is an information officer. Her family life in Alaska but her husband wants them to move. The preferred hobby for him and his kids is to perform lacross and he would never give it up.<br><br>My web page ... real psychic ([http://www.rosextv.com/uprofile.php?UID=389761 http://www.rosextv.com])

Functional equation (L-function) - Revision history

en>Bgwhite: WP:CHECKWIKI error fix. Broken bracket problem. Do general fixes and cleanup if needed. - using AWB (9507)

en>SD5bot: BOT: replacing template(s) per discussion about misleading template name