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By Jan-Hendrik Evertse

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In 1948, Erd˝os and Selberg independently found an elementary proof, “elementary” meaning that the proof avoids complex analysis or functional analysis, but definitely not that the proof is easy! In 1980, Newman gave a new, simple proof of the Prime Number 28 Theorem, based on complex analysis. Korevaar observed that Newman’s approach can be used to prove a simpler version of the Wiener-Ikehara Tauberian theorem with a not so difficult proof based on complex analysis alone and avoiding functional analysis.

2δe2πit )n We apply the Fubini-Tonelli Theorem (see the Prerequisites). Note that since |w − z| < δ, ∞ 1 D 0 f (x, z + 2δe2πit ) (w − z)n (2δe2πit )n n=0 1 ∞ M (x)2−n D 0 dt dx 2M (x)dx < ∞. dt dx D n=0 54 So the conditions of the Fubini-Tonelli Theorem are satisfied, and in the expression for F (w) derived above we can interchange the integrations and the summation. 10, we obtain ∞ 1 (w − z)n F (w) = n=0 D 0 D 1 2πi ∞ (w − z)n = n=0 ∞ (w − z)n = D n=0 f (x, z + 2δe2πit ) dt dx (2δe2πit )n γz,2δ f (x, ζ) · dζ (ζ − z)n+1 dx f (n) (x, z) · dx .

If F1 , F2 are any two analytic functions on U with F1 = F2 = f , then F1 − F2 is constant on U since U is connected. This shows that an anti-derivative of f is determined uniquely up to addition with a constant. It thus suffices to prove the existence of an analytic function F on U with F = f . 39 Fix z0 ∈ U . Given z ∈ U , we define F (z) by f (w)dw, F (z) := γz where γz is any path in U from z0 to z. This does not depend on the choice of γz . For let γ1 , γ2 be any two paths in U from z0 to z.

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