Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.4

Another Proof of Zagier's Matrix Conjecture

Yawen Ma and Lee-Peng Teo
Department of Mathematics
Xiamen University Malaysia
Jalan Sunsuria, Bandar Sunsuria
43900, Sepang, Selangor


We prove a conjecture of Zagier about the inverse of a $(K-1)\times (K-1)$ matrix $A=A_{K}$ using elementary methods. This formula allows one to express the product of single zeta values $\zeta(2r)\zeta(2K+1-2r)$ for $1\leq r\leq K-1$ in terms of the double zeta values $\zeta(2r,
2K+1-2r)$ for $1\leq r\leq K-1$ and $\zeta(2K+1)$.

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Received March 1 2022; revised versions received March 2 2022; June 28 2022; June 29 2022. Published in Journal of Integer Sequences, June 29 2022.

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