Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

Takashi Nakamura; Łukasz Pańkowski

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arXiv · 1212.5890

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

Takashi Nakamura·Łukasz Pańkowski

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Abstract

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler-Zagier-Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

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Takashi Nakamura, Łukasz Pańkowski. 2013-09-27. Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line. https://arxiv.org/abs/1212.5890

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