标题： 关于实线上具有相同参数的多重泽塔函数的行为 I 显示英文标题

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

摘要： 在目前这一系列论文中，我们研究了Euler-Zagier类型的r重ζ函数在实线上具有相同自变量时的行为。 在本文的第一部分，我们考虑该函数在区间[0,1]上的行为。 我们的基本工具是牛顿经典恒等式的“无穷”版本。 我们进行了数值计算，并绘制了实数s在[0,1]上，对于几个小的r值的图形。这些图形暗示了r重ζ函数的各种性质，其中一些我们已经严格证明。 例如，我们证明了r重ζ函数有r个渐近线，并确定了接近这些渐近线时的行为。 到目前为止，已知r=2时存在一个实零点。 我们的当前计算在r=3,...,10的情况下，在渐近线之间建立了多个新的实零点。 此外，关于实零点的数量，我们提出了一个猜想，并推导出一个用于计算区间[0,1]上零点数量的公式。 显示英文摘要

摘要： 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.