Effective Euler Sums
Abstract
Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo ζ(2)-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.
Related
- A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin
- Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin
- Effective estimates for exponential sums with multiplicative coefficients
- Sums of Kloosterman sums formed with modular symbols
- Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
BibTeX
@misc{zhao2026effective,
title = {{Effective Euler Sums}},
author = {Jianqiang Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09260},
eprint = {2609.09260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09260},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective Euler Sums
AU - Jianqiang Zhao
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09260
UR - https://arxiv.org/abs/2609.09260
AB - Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo ζ(2)-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.
ER -