On the behavior of multiple zeta-functions with identical arguments on the real line I

TL;DR

This paper investigates the behavior of Euler-Zagier multiple zeta-functions with identical arguments on the real line, providing new insights into their asymptotes, zeros, and conjectured properties through numerical and theoretical analysis.

Contribution

It introduces an infinite version of Newton's identities to analyze the zeta-functions, proves the existence of asymptotes and zeros, and formulates a conjecture on the zeros' distribution.

Findings

The r-fold zeta-function has r asymptotes.

Numerical computations reveal multiple real zeros between asymptotes for r=3,...,10.

A conjecture and a formula for the number of zeros on [0,1] are proposed.

Abstract

In the present series of papers, we study the behavior of the r-fold zeta-function of Euler-Zagier type with identical arguments on the real line. In this first part, we consider the behavior on the interval [0,1]. Our basic tool is an "infinite" version of Newton's classical identities. We carry out numerical computations, and draw graphs for real s in [0,1], for several small values of r. Those graphs suggest various properties of the r-fold zeta-function, some of which we prove rigorously. For example, we show that the r-fold zeta-function has r asymptotes, and determine the asymptotic behavior close to those asymptotes. Until now, the existence of one real zero for r=2 has been known. Our present computations establish several new real zeros between asymptotes in the cases r=3,...,10. Moreover, on the number of real zeros, we raise a conjecture, and a formula for calculating the number of zeros on the interval [0,1] is derived.