Lower-order terms in mean-square formulas for the Euler--Zagier double zeta-function and regularized multiple zeta values

Abstract

We study the mean square of the Euler--Zagier double zeta-function when the second variable moves vertically. Mean-square formulas in and beyond the region of absolute convergence were previously obtained by Matsumoto and Tsumura and by Ikeda, Matsuoka and Nagata. In particular, the latter authors determined the leading terms in the following three boundary cases: \[ ζ_2\left(1+a,\frac12+it\right),\qquad ζ_2\left(1+ib,\frac12+it\right),\qquad ζ_2\left(1-a,\frac12+a+it\right), \] where $a\in\mathbb C$ with $\Re a>0$ and $b\in\mathbb R$ are fixed. The mean squares in the first and third cases, as well as in the second case when $b\neq0$, have leading terms of order $T\log T$. When $b=0$, the second case reduces to the corner $(1,1/2)$, where the leading term is of order $T(\log T)^3$. In the present paper, we refine these results by obtaining asymptotic formulas with remainder $o(T)$. In the case $b\neq0$, an additional oscillatory term of order $T$ occurs. We also show that, at the parameter points $(s_1,σ_2)=(k,\ell/2)$ with integers $k\geq1$ and $\ell\geq2$, the constant multiplying $T$ in the mean-square formula is a linear combination of multiple zeta values. At the boundary points $(k,1/2)$ with integers $k\geq1$, the harmonic finite parts of the corresponding divergent sums are expressed in terms of harmonic regularized multiple zeta values.