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en>Hhhippo: /* Discussion */ resolving dablink

2014-04-16T20:27:18Z

Discussion: resolving dablink

← Older revision		Revision as of 22:27, 16 April 2014
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	~~In [[mathematics]], and more particularly in [[analytic number theory]]~~, '~~''Perron's formula'''~~ is a ~~formula due to [[Oskar Perron]] to calculate the sum of an arithmetical function, by means of an inverse [[Mellin transform]]~~.		Nice to meet you, my name is Numbers Held although I don't really like becoming called like that. My day job is a meter reader. For a while I've been in South Dakota and my mothers and fathers live nearby. Playing baseball is the hobby he will never quit performing.<br><br>my weblog [http://www.cam4teens.com/blog/84472 www.cam4teens.com]

	~~==Statement==~~
	~~Let <math>\{~~a~~(n)\}</math> be an [[arithmetic function]],~~ and ~~let~~

	~~:<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} </math>~~

	~~be the corresponding [[Dirichlet series]]. Presume the Dirichlet series to be [[absolutely convergent]] for <math>\Re(s)>\sigma_a</math>~~. ~~Then Perron's formula~~ is

	~~:<math> A(x) = {\sum_{n\le x}}^{\star} a(n)~~
	~~=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(z)\frac{x^{z}}{z} dz.\; </math>~~

	~~Here,~~ the ~~star on the summation indicates that the last term of the sum must be multiplied by 1/2 when ''x'' is an integer~~. ~~The formula requires <math>c>\sigma_a~~<~~/math~~> ~~and~~ <~~math>x~~>~~0</math> real, but otherwise arbitrary.~~

	~~==Proof==~~
	~~An easy sketch of the proof comes from taking~~ [~~[Abel's summation formula\|Abel's sum formula]]~~

	:~~<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{0}^{\infty} A(x)x^{-(s+1) } dx. <~~/~~math>~~

	~~This is nothing but a [[Laplace transform]] under the variable change <math>x=e^t.<~~/~~math> Inverting it one gets Perron's formula~~.

	~~==Examples==~~
	~~Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums~~. ~~Thus, for example, one has the famous integral representation for the [[Riemann zeta function]]:~~

	~~:<math>\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx</math>~~

	~~and a similar formula for [[Dirichlet L-function]]s:~~

	~~:<math>L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx<~~/~~math>~~

	~~where~~

	~~:<math>A(x)=\sum_{n\le x} \chi(n)<~~/~~math>~~

	~~and <math>\chi(n)</math> is a [[Dirichlet character]]~~. ~~Other examples appear in the articles on the [[Mertens function]] and the [[von Mangoldt function]]~~.

	~~== References ==~~
	* Page 243 of {{Apostol IANT}}
	* {{mathworld\|urlname=PerronsFormula\|title=Perron's formula}}
	*{{cite book \|last=Tenebaum \|first=Gérald \|year=1995 \|title=Introduction to analytic and probabilistic number theory \|publisher=Cambridge University Press \|location=Cambridge \|isbn=0-521-41261-7 }}

	~~[[Category:Analytic number theory]]~~
	~~[[Category:Calculus]]~~
	~~[[Category:Integral transforms]]~~
	~~[[Category:Summability methods]~~]

en>BD2412: minor fixes, mostly disambig links using AWB

2013-06-02T16:58:51Z

minor fixes, mostly disambig links using AWB

← Older revision		Revision as of 18:58, 2 June 2013
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	~~I am Oscar~~ and ~~I completely dig that title. His spouse doesn~~'~~t like it the way he does but what he truly likes performing~~ is to ~~do aerobics~~ and ~~he'~~s ~~been doing it~~ for ~~quite a while~~. ~~Minnesota is where he~~'s ~~been residing for years~~. ~~He utilized to~~ be ~~unemployed but now he~~ is ~~a computer operator~~ but ~~his marketing never~~ comes.<br><br>~~Feel free~~ to ~~surf~~ to ~~my website;~~ [~~http~~://~~www.animecontent.com~~/~~blog~~/~~461505 www~~.~~animecontent~~.~~com~~]		In [[mathematics]], and more particularly in [[analytic number theory]], '''Perron's formula''' is a formula due to [[Oskar Perron]] to calculate the sum of an arithmetical function, by means of an inverse [[Mellin transform]].

			==Statement==
			Let <math>\{a(n)\}</math> be an [[arithmetic function]], and let

			:<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} </math>

			be the corresponding [[Dirichlet series]]. Presume the Dirichlet series to be [[absolutely convergent]] for <math>\Re(s)>\sigma_a</math>. Then Perron's formula is

			:<math> A(x) = {\sum_{n\le x}}^{\star} a(n)
			=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(z)\frac{x^{z}}{z} dz.\; </math>

			Here, the star on the summation indicates that the last term of the sum must be multiplied by 1/2 when ''x'' is an integer. The formula requires <math>c>\sigma_a</math> and <math>x>0</math> real, but otherwise arbitrary.

			==Proof==
			An easy sketch of the proof comes from taking [[Abel's summation formula\|Abel's sum formula]]

			:<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{0}^{\infty} A(x)x^{-(s+1) } dx. </math>

			This is nothing but a [[Laplace transform]] under the variable change <math>x=e^t.</math> Inverting it one gets Perron's formula.

			==Examples==
			Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums. Thus, for example, one has the famous integral representation for the [[Riemann zeta function]]:

			:<math>\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx</math>

			and a similar formula for [[Dirichlet L-function]]s:

			:<math>L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx</math>

			where

			:<math>A(x)=\sum_{n\le x} \chi(n)</math>

			and <math>\chi(n)</math> is a [[Dirichlet character]]. Other examples appear in the articles on the [[Mertens function]] and the [[von Mangoldt function]].

			== References ==
			* Page 243 of {{Apostol IANT}}
			* {{mathworld\|urlname=PerronsFormula\|title=Perron's formula}}
			*{{cite book \|last=Tenebaum \|first=Gérald \|year=1995 \|title=Introduction to analytic and probabilistic number theory \|publisher=Cambridge University Press \|location=Cambridge \|isbn=0-521-41261-7 }}

			[[Category:Analytic number theory]]
			[[Category:Calculus]]
			[[Category:Integral transforms]]
			[[Category:Summability methods]]

en>SchreiberBike: Disambiguation needed for the link Basis

2012-04-24T04:19:11Z

Disambiguation needed for the link Basis

New page

I am Oscar and I completely dig that title. His spouse doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been doing it for quite a while. Minnesota is where he's been residing for years. He utilized to be unemployed but now he is a computer operator but his marketing never comes.<br><br>Feel free to surf to my website; [http://www.animecontent.com/blog/461505 www.animecontent.com]