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On spectral theory of the Riemann zeta function

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Abstract

Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-Pólya space. It has geometric multiplicity one. A relation between nontrivial zeros of the zeta function and eigenvalues of the convolution operator is given. It is an analogue of the Selberg transform in Selberg’s trace formula. Elements of the Hilbert-Pólya space are characterized by the Poisson summation formula.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Xian-Jin Li

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  1. Xian-Jin Li
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Correspondence to Xian-Jin Li.

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Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

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Li, XJ. On spectral theory of the Riemann zeta function. Sci. China Math. 62, 2317–2330 (2019). https://doi.org/10.1007/s11425-018-9356-0

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  • DOI: https://doi.org/10.1007/s11425-018-9356-0

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