论文标题
关于多个Zeta功能的行为，在真实行上具有相同的参数i
On the behavior of multiple zeta-functions with identical arguments on the real line I
论文作者
论文摘要
在当前的一系列论文中，我们研究了Euler-Zagier类型的R折Zeta功能的行为，并在真实线上具有相同的参数。在第一部分中，我们考虑在间隔[0,1]上的行为。我们的基本工具是牛顿经典身份的“无限”版本。我们执行数值计算，并在[0,1]中为r的几个小值绘制真实s的图。这些图提示了R折Zeta功能的各种特性，其中一些我们严格证明。例如，我们表明R折Zeta功能具有R渐近性，并确定接近这些渐近线的渐近行为。到目前为止，已知r = 2的一个真实零的存在。我们目前的计算在渐近线之间建立了几个新的实际零，在r = 3，...，10。此外，在实际零的数量上，我们提出了一个猜想，并得出了计算间隔[0,1]上零数的公式。
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.
