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| The '''Wiener–Ikehara theorem''' can be used to prove the [[prime number theorem]] (PNT) (Chandrasekharan, 1969). It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]] in 1932. It is an example of a [[Tauberian theorem]]. | | The writer is known as Wilber Pegues. Invoicing is my profession. For a whilst I've been in Alaska but I will have to transfer in a year or two. Doing ballet is something she would never give up.<br><br>My web site - are psychics real ([http://isaworld.pe.kr/?document_srl=392088 visit the following website page]) |
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| == Statement ==
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| Let ''A''(''x'') be a non-negative, [[monotonic function|monotonic]] nondecreasing function of ''x'', defined for 0 ≤ ''x'' < ∞. Suppose that
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| :<math>\int_0^\infty A(x) e^{-xs}\,dx</math>
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| converges for ℜ(''s'') > 1 to the function ''ƒ''(''s'') and that ''ƒ''(''s'') is [[analytic function|analytic]] for ℜ(''s'') ≥ 1, except for a simple [[Pole (complex analysis)|pole]] at ''s'' = 1 with [[Residue (complex analysis)|residue]] 1: that is,
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| :<math>f(s) - \frac{1}{s-1}</math>
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| is analytic in ℜ(''s'') ≥ 1. Then the [[Limit of a function|limit]] as ''x'' goes to infinity of ''e''<sup>−''x''</sup> ''A''(''x'') is equal to 1. | |
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| == Application ==
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| An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form
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| :<math>\sum_{n=1}^\infty a(n) n^{-s}</math> | |
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| where ''a''(''n'') is non-negative. If the series converges to an analytic function in
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| :<math>\Re(s) \ge b\,</math>
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| with a simple pole of residue ''c'' at ''s'' = ''b'', then
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| :<math>\sum_{n\le X}a(n) \sim \frac{c}{b} X^b.</math>
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| Applying this to the logarithmic derivative of the [[Riemann zeta function]], where the coefficients in the Dirichlet series are values of the [[von Mangoldt function]], it is possible to deduce the [[Prime number theorem|PNT]] from the fact that the zeta function has no zeroes on the line
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| :<math>\Re(s)=1. \, </math>
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| ==References==
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| *{{citation| author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=Journal of Mathematics and Physics of the Massachusetts Institute of Technology | year=1931 | volume=10 | pages=1–12 |zbl=0001.12902}}
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| *{{Citation | last1=Wiener | first1=Norbert | title=Tauberian Theorems | jstor=1968102 | series=Second Series | doi=10.2307/1968102 | jfm=
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| 58.0226.02 | year=1932 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=33 | issue=1 | pages=1–100}}
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| *{{cite book | author=K. Chandrasekharan | title=Introduction to Analytic Number Theory | series=Grundlehren der mathematischen Wissenschaften | publisher=[[Springer-Verlag]] | year=1969 | isbn=3-540-04141-9 }}
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| * {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=259–266 | publisher=Cambridge Univ. Press | location=Cambridge }}
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| {{DEFAULTSORT:Wiener-Ikehara theorem}}
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| [[Category:Number theory]]
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| [[Category:Tauberian theorems]]
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| [[Category:Theorems in analysis]]
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The writer is known as Wilber Pegues. Invoicing is my profession. For a whilst I've been in Alaska but I will have to transfer in a year or two. Doing ballet is something she would never give up.
My web site - are psychics real (visit the following website page)