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

TL;DR

This paper investigates the behavior of multiple zeta-functions with identical arguments on the real line, providing new asymptotic formulas, numerical evidence for zeros, and conjectures about their properties.

Contribution

It introduces an infinite version of Newton's identities, analyzes the asymptotic behavior near asymptotes, and conjectures the zero distribution of these functions for various r.

Findings

$ ext{zeta}_r(s, ext{...},s)$ has $r$ asymptotes at $ ext{Re}(s)=1/k$ for $1 extless k extless r$

Numerical computations reveal multiple real zeros for $2 extless r extless 10$

Proves asymptotic formulas for $ ext{zeta}_r(-k, ext{...},-k)$ for odd positive integers $k$

Abstract

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $\zeta_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical computations, and draw graphs of $\zeta_r(s,s,\ldots,s)$ for real $s$, for several small values of $r$. Those graphs suggest various properties of $\zeta_r(s,s,\ldots,s)$, some of which we prove rigorously. When $s \in [0,1]$, we show that $\zeta_r(s,s,\ldots,s)$ has $r$ asymptotes at $\Re s=1/k$ ($1\leq k\leq r$), and determine the asymptotic behavior of $\zeta_r(s,s,\ldots,s)$ close to those asymptotes. Numerical computations establish the existence of several real zeros for $2\leq r\leq 10$ (in which only the case $r=2$ was previously known). Based on those computations, we raise a conjecture on the number of zeros for general $r$, and gives a formula for calculating the number of zeros. We also consider the behavior of $\zeta_r(s,s,\ldots,s)$ outside the interval $[0,1]$. We prove asymptotic formulas for $\zeta_r(-k,-k,\ldots,-k)$, where $k$ takes odd positive integer values and tends to $+\infty$. Moreover, on the number of real zeros of $\zeta_r(s,s,\ldots,s)$, we prove that there are exactly $(r-1)$ real zeros on the interrval $(-2n,-2(n-1))$ for any $n \geq 2$.