We introduce new zeta functions related to an endomorphism ϕ of a discrete group Γ. They are of two types: counting numbers of fixed (ρ ~ ρ o ϕⁿ) irreducible representations for iterations of ϕ from an appropriate dual space of Γ and counting Reidemeister numbers R(φⁿ) of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases, it is proved that these zeta functions coincide. The Gauss congruences for coefficients are proved. Useful asymptotic formulas for the zeta functions are found. Rationality is proved for some classes of groups, including those, which give also the first counterexamples simultaneously for TBFT (R(ϕ) = the number of fixed irreducible unitary representations) and TBFTf (R(ϕ) = the number of fixed irreducible unitary finite-dimensional representations) for an automorphism ϕ with R(ϕ) < 8.

中文翻译：

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Rememeister类型的新Zeta函数和扭曲的Burnside-Frobenius理论

我们介绍了与离散组Γ的内态ϕ相关的新zeta函数。它们有两种类型：固定的（计数数ρ〜ρ ö φ ^Ñ），用于迭代不可约表示φ从Γ的适当偶空间和计数Reidemeister号码- [R （φ ^Ñ）的不同压实。获得了这些函数的许多特性及其系数。在许多情况下，证明了这些zeta函数是一致的。证明了系数的高斯同余。找到了有关zeta函数的有用渐近公式。证明了某些类别的群体的合理性，包括那些同时给出TBFT（R（ϕ）=固定不可约unit表示的数量）和TBFT f（R（ϕ）=固定不可约number的数量）的第一个反例。有限维表示），用于同构φ与[R （φ）<8。