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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Zeros of polynomials of derivatives of zeta functions
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by Takashi Nakamura PDF
Proc. Amer. Math. Soc. 145 (2017), 2849-2858 Request permission

Abstract:

Let $P_s \in \mathcal {D}_s[X_0,X_1, \ldots ,X_l]$ be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane $\Re (s) > 1/2$. In the present paper, we show that the function $P_s(L(s), L^{(1)}(s),\ldots , L^{(l)}(s))$ has infinitely many zeros in the vertical strip $D:= \{ s \in {\mathbb {C}} : 1/2 < \Re (s) <1\}$ if $L(s)$ is hybridly universal and $P_s \in \mathcal {D}_s[X_0,X_1, \ldots ,X_l]$ is a polynomial such that at least one of the degrees of $X_1,\ldots ,X_l$ is greater than zero. As a corollary, we prove that the function $(d^k / ds^k) P_s(L(s))$ with $k \in {\mathbb {N}}$ has infinitely many zeros in the strip $D$ when $L(s)$ is hybridly universal and $P_s \in \mathcal {D}_s[X]$ is a polynomial with degree greater than zero. The upper bounds for the numbers of zeros of $P_s(L(s), L^{(1)}(s),\ldots , L^{(l)}(s))$ and $(d^k / ds^k) P_s(L(s))$ are studied as well.
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Additional Information
  • Takashi Nakamura
  • Affiliation: Department of Liberal Arts, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken, 278-8510, Japan
  • MR Author ID: 755913
  • Email: nakamuratakashi@rs.tus.ac.jp
  • Received by editor(s): May 19, 2016
  • Received by editor(s) in revised form: August 22, 2016
  • Published electronically: January 25, 2017
  • Additional Notes: The author was partially supported by JSPS grant 16K05077 and Japan-France Research Cooperative Program (JSPS and CNRS).
  • Communicated by: Ken Ono
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2849-2858
  • MSC (2010): Primary 11M26
  • DOI: https://doi.org/10.1090/proc/13460
  • MathSciNet review: 3637935