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| − | =Wiener–Ikehara theorem=
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| − | A [[Tauberian theorem]] relating the behaviour of a real sequence to the analytic properties of the associated [[Dirichlet series]]. It is used in the study of [[arithmetic function]]s and yields a proof of the [[Prime number theorem]]. It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]] in 1932.
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| − | Let $F(x)$ be a non-negative, [[monotonic function|monotonic]] decreasing function of the positive [[real number|real]] variable $x$. Suppose that the [[Laplace transform]]
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| − | $$
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| − | \int_0^\infty F(x)\exp(-xs) dx
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| − | $$
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| − | converges for $\Re s >1$ to the function $f(s)$ and that $f(s)$ is [[analytic function|analytic]] for $\Re s \ge 1$, except for a simple [[pole]] at $s=1$ with residue 1. Then the [[Limit of a function|limit]] as $x$ goes to infinity of $e^{-x} F(x)$ is equal to 1.
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| − | An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-negative. If the series converges to an analytic function in $\Re s \ge b$ with a simple pole of residue $c$ at $s = b$, then $\sum_{n\le X}a(n) \sim c \cdot X^b$.
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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 from the fact that the zeta function has no zeroes on the line $\Re (s)=1$.
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| − | ==References==
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| − | *{{User:Richard Pinch/sandbox/Ref | author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=J. Math. Phys. | year=1931 | volume=10 | pages=1–12 }}
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| − | *{{User:Richard Pinch/sandbox/Ref | author=N. Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=[[Annals of Mathematics]] | year=1932 | volume=33 | pages=1–100 }}
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| − | *{{User:Richard Pinch/sandbox/Ref | 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 }}
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Revision as of 16:25, 7 September 2013
How to Cite This Entry:
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30405