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: ζ2(1+a,1/2+it), ζ2(1+ib,1/2+it), ζ2(1−a,1/2+a+it), where a∈ℂ with ℜ a>0 and b∈ℝ are fixed. The mean squares in the first and third cases, as well as in the second case when b≠0, have leading terms of order Tlog 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≠0, an additional oscillatory term of order T occurs. We also show that, at the parameter points (s1,σ2)=(k,ℓ/2) with integers k≥1 and ℓ≥2, 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≥1, the harmonic finite parts of the corresponding divergent sums are expressed in terms of harmonic regularized multiple zeta values.
Related
- Value-distribution of Euler--Zagier type multiple zeta-functions with identical arguments on the critical line
- Zagier's 2-3-2 formula for multiple Eisenstein series
- Multiple Zeta Values in Positive Characteristic: A Kaneko-Zagier-Type Conjecture
- Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
- Weighted sum formulas for finite multiple mixed values
BibTeX
@misc{onozuka2026lowerorder,
title = {{Lower-order terms in mean-square formulas for the Euler--Zagier double zeta-function and regularized multiple zeta values}},
author = {Tomokazu Onozuka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23413},
eprint = {2609.23413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23413},
}
RIS
TY - UNPB
TI - Lower-order terms in mean-square formulas for the Euler--Zagier double zeta-function and regularized multiple zeta values
AU - Tomokazu Onozuka
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.23413
UR - https://arxiv.org/abs/2609.23413
AB - 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(1+a,1/2+it), ζ_2(1+ib,1/2+it), ζ_2(1−a,1/2+a+it), where a∈ℂ with ℜ a>0 and b∈ℝ are fixed. The mean squares in the first and third cases, as well as in the second case when b≠0, have leading terms of order Tlog 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≠0, an additional oscillatory term of order T occurs. We also show that, at the parameter points (s_1,σ_2)=(k,ℓ/2) with integers k≥1 and ℓ≥2, 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≥1, the harmonic finite parts of the corresponding divergent sums are expressed in terms of harmonic regularized multiple zeta values.
ER -