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On the roots of the equation \(\zeta (s)=a\)

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Abstract

Given any complex number \(a\), we prove that there are infinitely many simple roots of the equation \(\zeta (s)=a\) with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve \(t\mapsto \zeta ({1\over 2}+it)\). Moreover, we show that the curve \(\mathbb {R}\ni t\mapsto (\zeta ({1\over 2}+it),\zeta '({1\over 2}+it))\) is not dense in \(\mathbb {C}^2\).

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Notes

  1. The paper [4] of Bohr, Landau and Littlewood consists of three independent chapters, the first belonging essentially to Bohr, the second to Landau, and the third to Littlewood.

  2. Recently, it was shown by Trudgian [27] that Gram’s law fails for a positive proportion. The first failure appears at \(t=282.454\).

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 , Vilnius, Lithuania

    Ramūnas Garunkštis

  2. Department of Mathematics, Würzburg University, Emil-Fischer-Str. 40, 97 218 , Würzburg, Germany

    Jörn Steuding

Authors
  1. Ramūnas Garunkštis
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  2. Jörn Steuding
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Correspondence to Ramūnas Garunkštis.

Additional information

Communicated by Ulf Kühn.

The first author is supported by Grant No MIP-94 from the Research Council of Lithuania.

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Garunkštis, R., Steuding, J. On the roots of the equation \(\zeta (s)=a\) . Abh. Math. Semin. Univ. Hambg. 84, 1–15 (2014). https://doi.org/10.1007/s12188-014-0093-7

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  • DOI: https://doi.org/10.1007/s12188-014-0093-7

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