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Linear operators, the Hurwitz zeta function and Dirichlet \(L\)-functions. (English) Zbl 1469.11338

J. Number Theory 217, 422-442 (2020); corrigendum ibid. 234, 499-502 (2022).
Summary: At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity [D. Hilbert, Arch. der Math. u. Phys. (3) 1, 44–63, 213–237 (1901; JFM 32.0084.05)]. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies [R. A. Van Gorder, J. Number Theory 147, 778–788 (2015; Zbl 1382.11060)]. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder’s operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder’s operator applied to the zeta function and show that a similar operator applied to zeta and other \(L\)-functions does converge.

MSC:

11M35 Hurwitz and Lerch zeta functions
30D15 Special classes of entire functions of one complex variable and growth estimates

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References:

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