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We construct explicit counterexamples that show that it is impossible to getany remainder, other than the classical ones, in the Wiener-Ikehara theorem andthe Ingham-Karamata theorem under just an additional analytic continuationhypothesis to a half-plane (or even to the whole complex plane).

On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach