Riemann bm$\zeta$-functions on semigroups

By introducing prime elements in semigroups, divisor functions and Mobius functions are defined and studied on a class of semigroups. The Riemann $\\zeta$-function can be naturally generalized through different means on these semigroups. The semigroup associated with Thompsons group is a typical example of our interest. We show that zeta functions on Thompsons semigroup obtained through different definitions agree with each other. Moreover, this zeta function has at least another real pole less than 1 besides a simple pole at 1. Many other arithmetic results are obtained on some noncommutative semigroups such as analogs of the prime number theorem on natural numbers.