Reducibility of Signed Cyclic Sums of Mordell-Tornheim Zeta and L-Values

Matsumoto et al. define the Mordell–Tornheim L-functions of depth k by

L_MT(s₁, . . . , s_k+1; χ₁, . . . ,χ_k+1)

for complex variables s₁, . . . , s_k+1 and primitive Dirichlet characters χ₁, . . . ,χ_k+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.