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

Takashi Nakamura, Łukasz Pańkowski|arXiv (Cornell University)|Dec 24, 2012

Analytic Number Theory Research32 references8 citations

TL;DR

This paper proves that any non-monomial polynomial of the Riemann zeta-function and related zeta-functions—such as those associated with symmetric matrices, spectral zeta-functions, Euler-Zagier multiple zeta-functions, Barnes and Shintani multiple zeta-functions, and Witten multiple zeta-functions—has infinitely many complex zeros off the critical line. The result is derived via hybrid universality and analytic continuation techniques, establishing a broad class of zeta-functions with pervasive non-trivial zeros beyond 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.

Motivation & Objective

• To establish a general criterion for the existence of infinitely many non-trivial complex zeros off the critical line in non-monomial polynomials of zeta-functions.
• To extend the universality principle to zeta-functions with zeros off the critical line, particularly through hybrid universality and analytic continuation.
• To resolve the distribution of zeros for multiple zeta-functions (e.g., Euler-Zagier, Barnes, Shintani, Witten) under algebraic or transcendental parameter conditions.
• To clarify the relationship between the Lindelöf hypothesis and the location of zeros, showing equivalence of the hypothesis for zeta-functions despite off-line zeros.

Proposed method

• Application of hybrid universality theorems to link approximation properties of zeta-functions with zero distribution.
• Use of meromorphic continuation via Euler-Maclaurin, Mellin-Barnes, and binomial theorem techniques for multiple zeta-functions.
• Analysis of the functional equation and non-vanishing properties of auxiliary functions to ensure non-trivial zero clusters.
• Leveraging the strong universality of zeta-functions with Euler products and their combinations with general Dirichlet series.
• Employment of Kronecker's approximation theorem in conjunction with spectral and representation-theoretic structures for Lie algebra-based zeta-functions.
• Proof by contradiction and density arguments to show that zero sets cannot be confined to the critical line under the stated conditions.

Experimental results

Research questions

• RQ1 Do non-monomial polynomials of the Riemann zeta-function have infinitely many complex zeros off the critical line?
• RQ2 Can hybrid universality theorems be used to deduce zero distribution theorems for zeta-functions with zeros off the critical line?
• RQ3 Do Barnes multiple zeta-functions with rational or transcendental parameters have infinitely many non-trivial zeros off the critical line?
• RQ4 Are the Mordell, Euler-Zagier-Hurwitz, and Witten multiple zeta-functions guaranteed to have infinitely many complex zeros off the critical line under certain parameter conditions?
• RQ5 Is the Lindelöf hypothesis for the Riemann zeta-function equivalent to the Lindelöf hypothesis for other zeta-functions that have zeros off the critical line?

Key findings

• Any non-monomial polynomial of the Riemann zeta-function has infinitely many complex zeros off the critical line, provided certain convergence and non-vanishing conditions hold.
• The zeta-functions associated to symmetric matrices, as studied by Ibukiyama and Saito, possess infinitely many non-trivial zeros off the critical line.
• Spectral zeta-functions and Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line when parameters are transcendental or algebraically independent.
• Barnes multiple zeta-functions with rational or transcendental parameters have infinitely many zeros off the critical line, implying the same for Shintani multiple zeta-functions under suitable conditions.
• The Mordell multiple zeta-functions, Euler-Zagier-Hurwitz type, and Witten multiple zeta-functions all have infinitely many complex zeros off the critical line when their parameters satisfy the required algebraic independence or transcendence conditions.
• The Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for all the zeta-functions discussed, despite the presence of zeros off the critical line.