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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]].
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== Statement  ==
Let ''A''(''x'') be a non-negative, [[monotonic function|monotonic]] nondecreasing function of ''x'', defined for 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;∞. Suppose that
 
:<math>\int_0^\infty A(x) e^{-xs}\,dx</math>
 
converges for ℜ(''s'')&nbsp;>&nbsp;1 to the function ''&fnof;''(''s'') and that ''&fnof;''(''s'') is [[analytic function|analytic]] for ℜ(''s'')&nbsp;≥&nbsp;1, except for a simple [[Pole (complex analysis)|pole]] at ''s''&nbsp;=&nbsp;1 with [[Residue (complex analysis)|residue]]&nbsp;1: that is,
 
:<math>f(s) - \frac{1}{s-1}</math>
 
is analytic in ℜ(''s'')&nbsp;≥&nbsp;1. Then the [[Limit of a function|limit]] as ''x'' goes to infinity of ''e''<sup>&minus;''x''</sup>&thinsp;''A''(''x'') is equal to&nbsp;1.
 
== Application ==
 
An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form
 
:<math>\sum_{n=1}^\infty a(n) n^{-s}</math>
 
where ''a''(''n'') is non-negative.  If the series converges to an analytic function in
 
:<math>\Re(s) \ge b\,</math>
 
with a simple pole of residue ''c'' at ''s''&nbsp;=&nbsp;''b'', then
 
:<math>\sum_{n\le X}a(n) \sim \frac{c}{b} X^b.</math>
 
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
 
:<math>\Re(s)=1. \, </math>
 
==References==
*{{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}}
*{{Citation | last1=Wiener | first1=Norbert | title=Tauberian Theorems | jstor=1968102 | series=Second Series | doi=10.2307/1968102 | jfm=
58.0226.02 | year=1932 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=33 | issue=1 | pages=1–100}}
*{{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 }}
* {{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 }}
 
{{DEFAULTSORT:Wiener-Ikehara theorem}}
[[Category:Number theory]]
[[Category:Tauberian theorems]]
[[Category:Theorems in analysis]]

Latest revision as of 16:27, 11 January 2015

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