In this article we define and study a zeta function \(\zeta _G\)—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value \(\zeta _G(k)^{-1}\) at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that \(\zeta _G\) is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of \(\zeta _G\). We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-\({\mathfrak {C}}\) groups, where \({\mathfrak {C}}\) is a class of finite groups with prescribed composition factors. We prove that every real number \(a \ge 1\) is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of \(\zeta _G\) are rational functions in \(p^{-s}\) if G is virtually abelian. For finite groups G we calculate \(\zeta _G\) using the rational representation theory of G.

在本文中，我们定义并研究了 zeta 函数\(\zeta _G\) — 类似于 Hasse-Weil zeta 函数 — 它枚举了（有限）群G的有限域上的绝对不可约表示。对于所有 UBE​​RG 群，此 Weil 表示 zeta 函数收敛于复数半平面，并允许欧拉积分解。我们进行这项研究的动机是观察到足够大的整数k处的倒数\(\zeta _G(k)^{-1}\)与k 个随机元素生成G的完整群环的概率一致。到目前为止获得的显式公式表明\(\zeta _G\)表现得相当好。本文的中心对象是Weil横坐标，即\(\zeta_G\)的收敛a ( G )的横坐标。我们计算自由阿贝尔群、自由阿贝尔prop、自由prop 、自由原幂和自由可溶群的Weil横坐标。更一般地，我们获得自由亲\({\mathfrak {C}}\)群的 Weil 横坐标的界限（有时是显式值），其中\({\mathfrak {C}}\)是一类有限具有规定组成因素的组。我们证明每个实数\(a \ge 1\)都是某个有限群G的韦尔横坐标a ( G ) 。此外，我们证明如果G实际上是阿贝尔函数，则\(\zeta _G\)的欧拉因子是\(p^{-s}\)中的有理函数。对于有限群G，我们使用G的有理表示理论计算\(\zeta _G\)。