PROOFS OF LUPU’S CONJECTURES FOR MULTIPLE ZETA VALUES AND MULTIPLE t -VALUES

Let r ≥ 1 r≥ 1 r greater than or equals 1 be an integer. For any multiple index s = ( s 1 , s 2 , … , s r ) ∈ Z ≥ 1 r s=(s₁,s₂,… ,sᵣ) ∈ Z≥ 1ʳ bold s equals left parenthesis s 1 comma s 2 comma ellipsis comma s Subscript r Baseline right parenthesis element of double struck upper Z Subscript greater than or equals 1 Superscript r with s r > 1 sᵣ > 1 s Subscript r Baseline greater than 1 , the multiple zeta value (MZV for short) is defined by ζ ( s 1 , s 2 , … , s r ) := ∑ 1 ≤ k 1 < k 2 < ⋯ < k r 1 k 1 s 1 k 2 s 2 ⋯ k r s r