A generating series is derived for the sum of multiple zeta values ending with a fixed string, implying the sum has depth bounded by the sum of the string entries.
The parity theorem for multiple polylogarithms
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abstract
We generalize the well-known parity theorem for multiple zeta values (MZV) to functional equations of multiple polylogarithms (MPL). This reproves the parity theorem for MZV with an additional integrality statement, and also provides parity theorems for special values of MPL at roots of unity (also known as coloured MZV). We give explicit formulas in depths 2 and 3 and provide a computer program to compute the functional equations.
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math.NT 1years
2026 1verdicts
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Multiple zeta values ending with a fixed string
A generating series is derived for the sum of multiple zeta values ending with a fixed string, implying the sum has depth bounded by the sum of the string entries.