Universality of $L$-Dirichlet functions and nontrivial zeros of the Riemann zeta-function

A. Laurinčikas   ; J. Petuškinaitė

Sbornik. Mathematics, Tome 210 (2019) no. 12, pp. 1753-1773

Résumé

We prove a joint discrete universality theorem for Dirichlet $L$-functions concerning joint approximation of a tuple of analytic functions by shifts $L(s+ih\gamma_k, \chi_1),\dots,L(s+ih\gamma_k,\chi_r)$, where $0<\gamma_1<\gamma_2<\dotsb$ is the sequence of imaginary parts of the nontrivial zeros of the Riemann zeta-function, $h$ is a fixed positive number, and $\chi_1,\dots,\chi_r$ are pairwise nonequivalent Dirichlet characters. We use a weak form of Montgomery's conjecture on the correlation of pairs of zeros of the Riemann zeta-function in the analysis. Moreover, we show the universality of certain compositions of Dirichlet $L$-functions with operators in the space of analytic functions. Bibliography: 31 titles.

Détail

Keywords: Montgomery's conjecture on correlation of pairs, Riemann zeta-function, Dirichlet $L$-function, nontrivial zeros, Voronin's theorem, universality.

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