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Zeros of polynomials of derivatives of zeta functions


Author: Takashi Nakamura
Journal: Proc. Amer. Math. Soc. 145 (2017), 2849-2858
MSC (2010): Primary 11M26
DOI: https://doi.org/10.1090/proc/13460
Published electronically: January 25, 2017
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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
Email: nakamuratakashi@rs.tus.ac.jp

DOI: https://doi.org/10.1090/proc/13460
Keywords: Hybrid universality, zeros of the derivatives of zeta functions
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
Article copyright: © Copyright 2017 American Mathematical Society