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

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.

17 pages. We changed the title and the organization

References in corpus (1)

• Mean value theorems for the double zeta-function

Cited by in corpus (5)

• Zeros of L-functions outside the critical strip
• Numerical computations on the zeros of the Euler double zeta-function I
• Zeros of polynomials of derivatives of zeta functions
• Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence
• Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions