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

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.