Sum formula for finite multiple zeta values. (English) Zbl 1329.11093

M. Kaneko [“Finite multiple zeta values \(\mod p\) and relations among multiple zeta values”, Sūrikaisekikenkyūsho Kōkyūroku 1813, 27–31 (2010)] conjectured that \[ \sum_{k_1+\cdots+ k_n=k,\,k_1\geq 2}\zeta_{\mathcal A}(k_1,\dots,k_n)=\left(1+(-1)^n\binom{k-1}{n-1}\right)\frac{B_{p-k}}{k}, \] and \[ \sum_{k_1+\cdots+ k_n=k,\,k_1\geq 2}\zeta_{\mathcal A}^{\star}(k_1,\dots,k_n)=\left((-1)^n+\binom{k-1}{n-1}\right)\frac{B_{p-k}}{k}, \] where \(\zeta_{\mathcal A}\) and \(\zeta_{\mathcal A}^{\star}\) are finite multiple zeta(-star) values.
In this paper, the authors prove these conjectures and extend the results for a wider class of summation domain. These are the finite versions of the corresponding well-known summation formulas of MZV theory.


11M32 Multiple Dirichlet series and zeta functions and multizeta values
Full Text: DOI arXiv Euclid


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