## Zeros of polynomials of derivatives of zeta functions

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**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.## References

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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