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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] |
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| ==Statement==
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| Let <math>\{a(n)\}</math> be an [[arithmetic function]], and let
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| :<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} </math>
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| 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
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| :<math> A(x) = {\sum_{n\le x}}^{\star} a(n)
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| =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(z)\frac{x^{z}}{z} dz.\; </math>
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| 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.
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| ==Proof==
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| An easy sketch of the proof comes from taking [[Abel's summation formula|Abel's sum formula]]
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| :<math> g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{0}^{\infty} A(x)x^{-(s+1) } dx. </math> | |
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| This is nothing but a [[Laplace transform]] under the variable change <math>x=e^t.</math> Inverting it one gets Perron's formula.
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| ==Examples==
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| 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]]:
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| :<math>\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx</math>
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| and a similar formula for [[Dirichlet L-function]]s:
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| :<math>L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx</math>
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| where
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| :<math>A(x)=\sum_{n\le x} \chi(n)</math>
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| and <math>\chi(n)</math> is a [[Dirichlet character]]. Other examples appear in the articles on the [[Mertens function]] and the [[von Mangoldt function]].
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| == References ==
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| * Page 243 of {{Apostol IANT}}
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| * {{mathworld|urlname=PerronsFormula|title=Perron's formula}}
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| *{{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 }}
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| [[Category:Analytic number theory]]
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| [[Category:Calculus]]
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| [[Category:Integral transforms]]
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| [[Category:Summability methods]]
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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.
my weblog www.cam4teens.com