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 tha ...

, originally published by

Shikao Ikehara was a Japanese mathematician. He was a student of Norbert Wiener at MIT (PhD 1930). Career Following Wiener in 1928, in 1931 Ikehara used Wiener's Tauberian theory to derive another proof of the prime number theorem, demonstrated solely via the ...

, a student of

Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...

's, in 1931. It is a special case of

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 L^1 or L^2 can be approximated by linear combi ...

s, which were published by Wiener one year later. It can be used to prove the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 p ...

(Chandrasekharan, 1969), under the assumption that 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 and its analytic c ...

has no zeros on the line of real part one.

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 ord ...

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 small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

for ℜ(''s'') ≥ 1. Then the limit 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 anal ...

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

, 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 Mang ...

, it is possible to deduce the

Prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 p ...

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