Distributional Wiener-Ikehara theorem and twin primes
Let \(S(t)\) be a non-decreasing, right-continuous function, supported on \([0, \infty)\), and define its Laplace transform \[ f(s) = s \int_0^{\infty} S(t)e^{-st} \, dt, \qquad s = \sigma + it. \] Suppose that \(f(s)\) is analytic in \(\text{Re}(s) > 1\) and that the function \[ g(s) = f(s) - A(s - 1)^{-1} \] has a limit \(g(1 + it)\) as \(\sigma \downarrow 1\), either in \(C_{\text{loc}}(\mathbb R)\) or in \(L^1_{\text{loc}}(\mathbb R)\). Under these hypotheses, the classical Wiener-Ikehara theorem yields \[ e^{-t}S(t) \to A \qquad \text{as}\quad t \to \infty. \tag \(*\) \] The author generalizes the Wiener-Ikehara theorem. With \(S(t)\) and \(f(s)\) as above, set \[ G(s) = s^{-1}f(s) - A(s - 1)^{-1}. \] The main result of the paper states that the limit relation (\(*\)) is equivalent to the distributional convergence of \(G(\sigma + it)\) as \(\sigma \downarrow 1\) to a pseudofunction \(G(1 + it)\). (A pseudofunction is the distributional Fourier transform of a bounded function which tends to zero at \(\pm\infty\).) As a corollary, the author shows that (a strong form of) the twin-prime conjecture is equivalent to a pseudofunction boundary behavior for a certain Dirichlet series.
scientific article; zbMATH DE number 2175007
scientific article