Tokyo Journal of Mathematics

On the Parity Conjecture for Multiple $L$-values of Conductor Four

Hirofumi TSUMURA

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In this paper, we prove that the multiple $L$-value of conductor $4$ can be expressed in terms of lower depth multiple $L$-values under the condition on the parity of its depth and weight. This can be regarded as a character analogue of what is called the "parity result" for multiple zeta values which was proved by Zagier.

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Tokyo J. Math., Volume 30, Number 1 (2007), 21-40.

First available in Project Euclid: 20 July 2007

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TSUMURA, Hirofumi. On the Parity Conjecture for Multiple $L$-values of Conductor Four. Tokyo J. Math. 30 (2007), no. 1, 21--40. doi:10.3836/tjm/1184963645.

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  • T. Arakawa and M. Kaneko, On multiple $L$-values, J. Math. Soc. Japan 56 (2004), 967–992.
  • J. M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electronic J. Combin. 3 (1996), $\sharp$R23.
  • K. Dilcher, Zeros of Bernoulli, generalized Bernoulli and Euler polynomials, Memoirs of Amer. Math. Soc. 386 (1988).
  • K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math. 142 (2006), 307–338.
  • D. Terhune, Evaluations of double $L$-values, J. Number Theory 105 (2004), 275–301.
  • H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc. 131 (2003), 3633–3641.
  • H. Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arith. 111 (2004), 27–42.
  • H. Tsumura, Multiple harmonic series related to multiple Euler numbers, J. Number Theory 106 (2004), 155–168.