A continuous version of multiple zeta functions and multiple zeta values (Q2687980)

In this paper, the author defines a continuous version of multiple zeta functions by \[ \zeta^{\mathscr C}(s_1,\ldots,s_r)=\int_1^\infty\cdots\int_1^\infty\frac{dx_1\cdots dx_r}{x_1^{s_1}(x_1+x_2)^{s_2}\cdot(x_1+\cdot+x_r)^{s_r}} \] and proves that these functions can be analytically continued to meromorphic functions on \(\mathbb C^r\) with only simple poles at some special hyperplanes. As in the classical multiple zeta values, it is shown that the values of these functions at positive integers, called \textit{continuous multiple zeta values}, satisfy the shuffle product and the sum formulas. The author also considers the depth defect phenomena in the algebra of continuous multiple zeta values, and the relation between continuous multiple zeta values and multiple polylogarithms.