Wiener–Ikehara theorem
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The Wiener–Ikehara theorem is a
Tauberian theorem In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that ...
introduced by . It follows from
Wiener's Tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations ...
, and can be used to prove the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
(Chandrasekharan, 1969).


Statement

Let ''A''(''x'') be a non-negative,
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
nondecreasing function of ''x'', defined for 0 ≤ ''x'' < ∞. Suppose that :f(s)=\int_0^\infty A(x) e^\,dx converges for ℜ(''s'') > 1 to the function ''ƒ''(''s'') and that, for some non-negative number ''c'', :f(s) - \frac has an extension as a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
for ℜ(''s'') ≥ 1. Then the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to c.


One Particular Application

An important number-theoretic application of the theorem is to
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...
of the form :\sum_^\infty a(n) n^ 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_a(n) \sim \frac X^b. Applying this to the logarithmic derivative of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, where the coefficients in the Dirichlet series are values of the
von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangold ...
, it is possible to deduce the
Prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
from the fact that the zeta function has no zeroes on the line :\Re(s)=1.


References

* * * * {{DEFAULTSORT:Wiener-Ikehara theorem Theorems in number theory Tauberian theorems