非交换半群上的ζ-函数

Riemann ζ-functions on semigroups

自然数是由素数生成的乘法半群,从推广素数乘积的非交换性得到一类具有算术性质的非交换半群,自然数上的M¨obius函数和Riemannζ-函数等得到了自然推广.经典的Thompson群的生成半群等例子都是我们研究的特殊情形,它们上面的ζ-函数和经典的ζ-函数有类似的性质,但也有本质差别.本文证明类似的素数定理对许多非交换算术半群成立.而Thompson半群的ζ-函数至少有两个极点,这种现象反映了非交换半群中因子分解的复杂性.

By introducing prime elements in semigroups, divisor functions and M¨obius functions are defined and studied on a class of semigroups. The Riemann ζ-function can be naturally generalized through different means on these semigroups. The semigroup associated with Thompson’s group is a typical example of our interest.We show that zeta functions on Thompson’s 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.