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

Abstract

We study the behavior of r-fold zeta-functions of Euler-Zagier type with identical arguments ζr(s,s,…,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 ζr(s,s,…,s) for real s, for several small values of r. Those graphs suggest various properties of ζr(s,s,…,s), some of which we prove rigorously. When s∈[0,1], we show that ζr(s,s,…,s) has r asymptotes at ℜs=1/k (1≤k≤r), and determine the asymptotic behavior of ζr(s,s,…,s) close to those asymptotes. Numerical computations establish the existence of several real zeros for 2≤r≤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 ζr(s,s,…,s) outside the interval [0,1]. We prove asymptotic formulas for ζr(−k,−k,…,−k), where k takes odd positive integer values and tends to +∞. Moreover, on the number of real zeros of ζr(s,s,…,s), we prove that there are exactly (r−1) real zeros on the interval (−2n,−2(n−1)) for any n≥2.