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The purpose of this paper is two-fold. First, we consider the classicalMordell--Tornheim zeta values and their alternating version. It is well-knownthat these values can be expressed as rational linear combinations of multiplezeta values (MZVs) and the alternating MZVs, respectively. We show that,however, the spaces generated by these values over the rational numbers are ingeneral much smaller than the MZV space and the alternating MZV space,respectively, which disproves a conjecture of Bachmann, Takeyama and Tasaka.Second, we study supercongruences of some finite sums of multiple integervariables. This kind of congruences is a variation of the so called finitemultiple zeta values when the moduli are primes instead of prime powers. Ingeneral, these objects can be transformed to finite analogs of theMordell--Tornheim sums which can be reduced to multiple harmonic sums. Thisapproach not only simplifies the proof of a few previous results but alsogeneralizes some of them. At the end of the paper, we also provide a conjecturesupported by strong numerical evidence.

Mordell--Tornheim Zeta Values, Their Alternating Version, and Their Finite Analogs