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

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments ${\zeta }_r(s,s,\dots ,s)$ 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 ${\zeta }_r(s,s,\dots ,s)$ for real $s$, for several small values of $r$. Those graphs suggest various properties of ${\zeta }_r(s,s,\dots ,s)$, some of which we prove rigorously. When $s\in [0,1]$, we show that ${\zeta }_r(s,s,\dots ,s)$ has $r$ asymptotes at $\mathrm{\Re }s=1\mathrm{/}k$ ($1\leq k\leq r$),anddeterminetheasymptoticbehaviorof$\zeta_r(s,s,\ldots,s)$ close to those asymptotes. Numerical computations establish the existence of several real zeros for $2\le r\le 10$ (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 ${\zeta }_r(s,s,\dots ,s)$ outside the interval $[0,1]$. We prove asymptotic formulas for ${\zeta }_r(-k,-k,\dots ,-k)$, where $k$ takes odd positive integer values and tends to $+\mathrm{\infty }$. Moreover, on the number of real zeros of ${\zeta }_r(s,s,\dots ,s)$, we prove that there are exactly $(r-1)$ real zeros on the interrval $(-2n,-2(n-1))$ for any $n\ge 2$.