Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line
Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line
复制标题
黎曼 zeta 函数的任何非单项多项式都具有超出临界线的复零点
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Lukasz Pa'nkowski
中科院分区：
文献类型：
--
作者：
Takashi Nakamura;Lukasz Pa'nkowski
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 Lindel\"of hypothesis for the Riemann zeta-function is equivalent to the Lindel\"of 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.
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 Lindel\"of hypothesis for the Riemann zeta-function is equivalent to the Lindel\"of 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.
双 zeta 函数的函数方程
DOI：
--
发表时间：
2004
期刊：
Math.Proc.Cambridge Phil.Soc. 136
影响因子：
--
作者：
Kajiwara;Takeshi;kato;Kazuya;K.Matsumoto
通讯作者：
K.Matsumoto