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Zhou, Xia
- Reducibility of Signed Cyclic Sums of Mordell-Tornheim Zeta and L-Values
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LMT(s1, . . . , sk+1; χ1, . . . ,χk+1)
for complex variables s1, . . . , sk+1 and primitive Dirichlet characters χ1, . . . ,χk+1. In this paper, we shall show that certain signed cyclic sums of Mordell-Tornheim L-values are rational linear combinations of products of multiple L-values of lower depths (i.e., reducible). This simultaneously generalizes some results of Subbarao and Sitaramachandrarao, Onodera, and Matsumoto et al. As a direct corollary, if the weight and depth of a Mordell-Tornheim zeta value have different parity then we can explicitly express it as a rational linear combinations of products of Mordell-Tornheim zeta values of lower depths. Another corollary is that for any positive integer n and integer k≥2, the Mordell–Tornheim sums ζMT({n}k, n) is reducible where {n}k denotes the string (n, . . . , n) with n repeating k times.
Authors
Jianqiang Zhao
1,
Xia Zhou
2
Affiliations
1 Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, US
2 Department of Mathematics, Zhejiang University, Hangzhou-310027, CN
1 Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, US
2 Department of Mathematics, Zhejiang University, Hangzhou-310027, CN
Source
Journal of the Ramanujan Mathematical Society, Vol 26, No 4 (2011), Pagination: 383-414Abstract
Matsumoto et al. define the Mordell–Tornheim L-functions of depth k byLMT(s1, . . . , sk+1; χ1, . . . ,χk+1)
for complex variables s1, . . . , sk+1 and primitive Dirichlet characters χ1, . . . ,χk+1. In this paper, we shall show that certain signed cyclic sums of Mordell-Tornheim L-values are rational linear combinations of products of multiple L-values of lower depths (i.e., reducible). This simultaneously generalizes some results of Subbarao and Sitaramachandrarao, Onodera, and Matsumoto et al. As a direct corollary, if the weight and depth of a Mordell-Tornheim zeta value have different parity then we can explicitly express it as a rational linear combinations of products of Mordell-Tornheim zeta values of lower depths. Another corollary is that for any positive integer n and integer k≥2, the Mordell–Tornheim sums ζMT({n}k, n) is reducible where {n}k denotes the string (n, . . . , n) with n repeating k times.