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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The first author was partially supported by JSPS Grants 21740024. The second author was partially supported by the Grant No. N N201 6059 40 from National Science Centre.
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Nakamura, T., Pańkowski, Ł. On complex zeros off the critical line for non-monomial polynomial of zeta-functions. Math. Z. 284, 23–39 (2016). https://doi.org/10.1007/s00209-016-1643-8
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DOI: https://doi.org/10.1007/s00209-016-1643-8
Keywords
- Hybrid universality
- Lindelöf hypothesis
- Zeros of zeta-functions associated to symmetric matrices
- Euler–Zagier multiple zeta-functions
- Spectral zeta-functions
- Barnes multiple zeta-functions
- Shintani multiple zeta-functions