Wiener–Ikehara theorem

The Wiener–Ikehara theorem is a Tauberian theorem introduced by Shikao Ikehara (1931). It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (PNT) (Chandrasekharan, 1969).

StatementEdit

Let A(x) be a non-negative, monotonic nondecreasing function of x, defined for 0 ≤ x < ∞. Suppose that

 

converges for ℜ(s) > 1 to the function ƒ(s) and that, for some non-negative number c,

 

has an extension as a continuous function for ℜ(s) ≥ 1. Then the limit as x goes to infinity of exA(x) is equal to c.

One Particular ApplicationEdit

An important number-theoretic application of the theorem is to Dirichlet series of the form

 

where a(n) is non-negative. If the series converges to an analytic function in

 

with a simple pole of residue c at s = b, then

 

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 PNT from the fact that the zeta function has no zeroes on the line

 

ReferencesEdit

  • S. Ikehara (1931), "An extension of Landau's theorem in the analytic theory of numbers", Journal of Mathematics and Physics of the Massachusetts Institute of Technology, 10: 1–12, Zbl 0001.12902
  • Wiener, Norbert (1932), "Tauberian Theorems", Annals of Mathematics, Second Series, 33 (1): 1–100, doi:10.2307/1968102, ISSN 0003-486X, JFM 58.0226.02, JSTOR 1968102
  • K. Chandrasekharan (1969). Introduction to Analytic Number Theory. Grundlehren der mathematischen Wissenschaften. Springer-Verlag. ISBN 3-540-04141-9.
  • Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. Cambridge: Cambridge Univ. Press. pp. 259–266. ISBN 0-521-84903-9.