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2013-09-16T14:37:59Z

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	~~If you present photography effectively~~, ~~it helps you look much~~ more ~~properly at the globe around you. It~~ is ~~thus, on these grounds that compel various web service provider companies~~ to ~~integrate~~ the ~~same in their packages too~~. * A community forum for debate of the ~~product together with some other customers in~~ the ~~comments spot. In the recent years~~, ~~there has been~~ a ~~notable rise in the number of companies hiring Indian Word~~ - ~~Press developers. Over~~ a ~~million people are using Wordpress to blog~~ and the number ~~of Wordpress users is increasing every day.~~ <br><br>		{{DISPLAYTITLE:''p''-adic ''L''-function}}
			In [[mathematics]], a '''''p''-adic zeta function''', or more generally a '''''p''-adic ''L''-function''', is a function analogous to the [[Riemann zeta function]], or more general [[L-function\|''L''-functions]], but whose [[domain of a function\|domain]] and [[codomain\|target]] are ''p-adic'' (where ''p'' is a [[prime number]]). For example, the domain could be the [[p-adic integer\|''p''-adic integers]] '''Z'''<sub>''p''</sub>, a [[profinite group\|profinite ''p''-group]], or a ''p''-adic family of [[Galois representation]]s, and the image could be the [[p-adic number\|''p''-adic numbers]] '''Q'''<sub>''p''</sub> or its [[algebraic closure]].

	~~In case you loved this article~~ and ~~you would want to receive details relating to~~ [~~http://dinky.in/?WordpressBackup78077 wordpress dropbox backup~~] ~~kindly visit~~ the ~~web site. Most Word~~ - ~~Press web developers can provide quality CMS website solutions and they price their services at reasonable rates~~. ~~You do not catch a user~~'s ~~attention through big and large pictures that usually takes a millennium~~ to ~~load up. This plugin allows~~ a ~~blogger get more Facebook fans on the related fan page. E~~-~~commerce websites are meant to be buzzed with fresh contents, graphical enhancements~~, ~~and functionalities. You can also get a free keyword tool that is to determine how strong other competing sites are and number of~~ the ~~searches on the most popular search sites.~~ <br><br>~~Photography is an entire activity in itself~~, ~~and a thorough discovery~~ of ~~it is beyond~~ the ~~opportunity of~~ this ~~content~~. ~~It was also the very~~ first ~~year that~~ the ~~category~~ of ~~Martial Arts was included in the Parents - Connect nationwide online poll~~, ~~allowing parents to vote for their favorite San Antonio Martial Arts Academy~~. ~~I hope~~ this ~~short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo~~ is ~~Scam or~~ a ~~Genuine. User friendly features~~ and ~~flexibility~~ that ~~Word~~ - ~~Press has to offer is second to none~~. ~~For any web design~~ and ~~development assignment~~, ~~this~~ is ~~definitely one~~ of the ~~key concerns, specifically for online retail outlets as well as e~~-~~commerce websites~~. <br><br>The ~~disadvantage~~ is ~~it requires~~ a ~~considerable amount of time to set every thing up. This plugin allows~~ a ~~webmaster to create complex layouts without having to waste so much time~~ with ~~short codes~~. ~~A higher percentage of women are marrying~~ at ~~older ages~~,~~many are delaying childbearing until their careers are established~~, the ~~divorce rate is high and many couples remarry and desire their own children. IVF~~ ,~~fertility~~,~~infertility expert,surrogacy specialist in India at Rotundaivf~~. ~~If your blog employs~~ the ~~permalink~~ function, ~~This gives your SEO efforts a boost~~, ~~and your visitors will know firsthand what~~'~~s in~~ the ~~post when seeing~~ the ~~URL~~. <br><br>~~Someone with~~ a ~~basic knowledge~~ of ~~setting up a website should be able to complete~~ the ~~process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online~~. ~~It can run~~ as ~~plugin and you can still get to that whole database just in circumstance your webhost does not have~~ a c - ~~Panel area. Word~~ - ~~Press~~ can also be ~~quickly extended however improvement API is not~~ as ~~potent~~ as ~~Joomla~~'s~~. with posts or testimonials updated as they are uploaded~~ to ~~a particular section~~ of the ~~website~~. ~~Verify whether your company has a team~~ of ~~developers or programmers having hands~~-on ~~experience~~ and ~~knowledge about all Word~~ - ~~Press concepts~~.		The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which [[Tomio Kubota]] and [[Heinrich-Wolfgang Leopoldt]] gave the first construction of a ''p''-adic ''L''-function {{harv\|Kubota\|Leopoldt\|1964}}—is via the ''p''-adic interpolation of [[special values of L-functions\|special values of ''L''-functions]]. For example, Kubota–Leopoldt used [[Kummer's congruence]]s for [[Bernoulli number]]s to construct a ''p''-adic ''L''-function, the '''''p''-adic Riemann zeta function''' ζ<sub>''p''</sub>(''s''), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). ''p''-adic ''L''-functions arising in this fashion are typically referred to as '''analytic ''p''-adic ''L''-functions'''. The other major source of ''p''-adic ''L''-functions—first discovered by [[Kenkichi Iwasawa]]—is from the arithmetic of [[cyclotomic field]]s, or more generally, certain [[Galois module]]s over [[tower of fields\|towers of cyclotomic fields]] or even more general towers. A ''p''-adic ''L''-function arising in this way is typically called an '''arithmetic ''p''-adic ''L''-function''' as it encodes arithmetic data of the Galois module involved. The [[main conjecture of Iwasawa theory]] (now a theorem due to [[Barry Mazur]] and [[Andrew Wiles]]) is the statement that the Kubota–Leopoldt ''p''-adic ''L''-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic ''p''-adic ''L''-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of ''L''-functions contain arithmetic information.

			==Dirichlet L-functions==

			The Dirichlet ''L''-function is given by the analytic continuation of
			:<math>L(s,\chi) = \sum_n\frac{\chi(n)}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}}</math>
			The Dirichlet ''L''-function at negative integers is given by
			:<math>L(1-n, \chi) = -\frac{B_{n,\chi}}{n}</math>
			where ''B''<sub>''n'',χ</sub> is a [[generalized Bernoulli number]] defined by
			:<math> \displaystyle \sum_{n=0}^\infty B_{n,\chi}\frac{t^n}{n!} = \sum_{a=1}^f\frac{\chi(a)te^{at}}{e^{ft}-1}</math>
			for χ a Dirichlet character with conductor ''f''.

			==Definition using interpolation==

			The Kubota–Leopoldt ''p''-adic ''L''-function ''L''<sub>''p''</sub>(''s'', χ) interpolates the Dirichlet ''L''-function with the Euler factor at ''p'' removed.
			More precisely, ''L''<sub>''p''</sub>(''s'', χ) is the unique continuous function of the ''p''-adic number ''s'' such that
			:<math> \displaystyle L_p(1-n, \chi) = (1-\chi(p)p^{n-1})L(1-n, \chi)</math>
			for positive integers ''n'' divisible by ''p'' − 1. The right hand side is just the usual Dirichlet ''L''-function, except that the Euler factor at ''p'' is removed, otherwise it would not be ''p''-adically continuous. The continuity of the right hand side is closely related to the [[Kummer congruence]]s.

			When ''n'' is not divisible by ''p'' − 1 this does not usually hold; instead
			:<math> \displaystyle L_p(1-n, \chi) = (1-\chi\omega^{-n}(p)p^{n-1})L(1-n, \chi\omega^{-n})</math>
			for positive integers ''n''.
			Here χ is twisted by a power of the [[Teichmuller character]] ω.

			==Viewed as a ''p''-adic measure==

			''p''-adic ''L''-functions can also be thought of as [[p-adic measure\|''p''-adic measures]] (or [[p-adic distribution\|''p''-adic distributions]]) on ''p''-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (as '''Q'''<sub>''p''</sub>-valued functions on '''Z'''<sub>''p''</sub>) is via the [[Mazur–Mellin transform]] (and [[class field theory]]).

			==Totally real fields==

			{{harvtxt\|Deligne\|Ribet\|1980}}, building upon previous work of {{harvtxt\|Serre\|1973}}, constructed analytic ''p''-adic ''L''-functions for totally real fields. Independently, {{harvtxt\|Barsky\|1978}} and {{harvtxt\|Cassou-Noguès\|1979}} did the same, but their approaches followed Takuro Shintani's approach to the study of the ''L''-values.

			==References==

			*{{Citation \| last1=Barsky \| first1=Daniel \| editor1-last=Amice \| editor1-first=Y. \| editor2-last=Barskey \| editor2-first=D. \| editor3-last=Robba \| editor3-first=P. \| title=Groupe d'Etude d'Analyse Ultramétrique (5e année: 1977/78) \| url=http://www.numdam.org/item?id=GAU_1977-1978__5__A9_0 \| publisher=Secrétariat Math. \| location=Paris \| isbn=978-2-85926-266-2 \| mr=525346 \| year=1978 \| volume=16 \| chapter=Fonctions zeta p-adiques d'une classe de rayon des corps de nombres totalement réels}}
			*{{Citation \| last1=Cassou-Noguès \| first1=Pierrette \| title=Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques \| doi=10.1007/BF01389911 \| mr=524276 \| year=1979 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=51 \| issue=1 \| pages=29–59}}
			*{{Citation \| last1=Coates \| first1=John \| title=On p-adic L-functions \| url=http://www.numdam.org/item?id=SB_1988-1989__31__33_0 \| mr=1040567 \| year=1989 \| journal=Astérisque \| issn=0303-1179 \| issue=177 \| pages=33–59}}
			*{{Citation \| last1=Colmez \| first1=Pierre \| title=Fontaine's rings and p-adic L-functions \| url=http://www.math.jussieu.fr/~colmez/tsinghua.pdf \| year=2004}}
			*{{Citation \| last1=Deligne \| first1=Pierre \| author1-link=Pierre Deligne \| last2=Ribet \| first2=Kenneth A. \| title=Values of abelian L-functions at negative integers over totally real fields \| doi=10.1007/BF01453237 \| mr=579702 \| year=1980 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=59 \| issue=3 \| pages=227–286}}
			*{{Citation \| last1=Iwasawa \| first1=Kenkichi \| title=On p-adic L-functions \| jstor=1970817 \| mr=0269627 \| year=1969 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=89 \| pages=198–205 \| doi=10.2307/1970817 \| issue=1 \| publisher=Annals of Mathematics}}
			*{{Citation \| last1=Iwasawa \| first1=Kenkichi \| title=Lectures on p-adic L-functions \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08112-0 \| mr=0360526 \| year=1972}}
			*{{Citation \| last1=Katz \| first1=Nicholas M. \| title=Algebraic geometry \|series=Proc. Sympos. Pure Math., \|volume= 29 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| mr=0432649 \| year=1975 \| chapter=p-adic L-functions via moduli of elliptic curves \| pages=479–506}}
			*{{Citation \| last1=Koblitz \| first1=Neal \| author1-link=Neal Koblitz \| title=p-adic Numbers, p-adic Analysis, and Zeta-Functions \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics, vol. 58 \| isbn=978-0-387-96017-3 \| mr=754003 \| year=1984}}
			*{{Citation \| last1=Kubota \| first1=Tomio \| last2=Leopoldt \| first2=Heinrich-Wolfgang \| title=Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen \| url= http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002180626 \| mr=0163900 \| year=1964 \| journal=[[Journal für die reine und angewandte Mathematik]] \| issn=0075-4102 \| volume=214/215 \| pages=328–339
			\|author1-link=Tomio Kubota\|author2-link=Heinrich-Wolfgang Leopoldt}}
			*{{Citation \| last1=Serre \| first1=Jean-Pierre \| author1-link=Jean-Pierre Serre \| editor1-last=Kuyk \| editor1-first=Willem \| editor2-last=Serre \| editor2-first=Jean-Pierre \| editor2-link=Jean-Pierre Serre \| title=Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series= Lecture Notes in Math \| isbn=978-3-540-06483-1 \| doi=10.1007/978-3-540-37802-0_4 \| mr=0404145 \| year=1973 \| volume=350 \| chapter=Formes modulaires et fonctions zêta p-adiques \| pages=191–268}}

			[[Category:Zeta and L-functions]]

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