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Wiener–Ikehara theorem : ウィキペディア英語版
Wiener–Ikehara theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (PNT) (Chandrasekharan, 1969).
== Statement ==
Let ''A''(''x'') be a non-negative, monotonic nondecreasing function of ''x'', defined for 0 ≤ ''x'' < ∞. Suppose that
:\int_0^\infty A(x) e^\,dx
converges for ℜ(''s'') > 1 to the function ''ƒ''(''s'') and that ''ƒ''(''s'') is analytic for ℜ(''s'') ≥ 1, except for a simple pole at ''s'' = 1 with residue 1: that is,
:f(s) - \frac
is analytic in ℜ(''s'') ≥ 1. Then the limit as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to 1.

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