Some Probabilistic Problems On Prime Polynomials On Finite Fields

Random ideas and techniques have penetrated into every corner of science,such as integrable systems,random geometry,random walks on the group,statistical physics,random algorithms,life sciences and so on.On the other hand,the polynomial theory on the finite field is very important for the study of its algebraic structure,and its applications axe quite wide,such as information security and coding theory,etc.This paper applies random ide.as to polynomial rings over finite.fields,and study some.probabilistic problems about prime polynomials.The second chapter is to discuss the q-zeta function on the polynomial ring.The Euler formula,which is similar to the Riemann(-function,is proved by probabilistic method;the probability of two polynomial co-prime are obtained by the q-zeta function;the statis-tical properties of polynomial factorization are shown that the number of factors are the geometric distribution.The third chapter mainly aims to study the statistical properties of the number of polynomial prime factors on the polynomial ring.The famous Erdos-Kac theorem(1940)in the number theory indicates that the number of a natural number ?(n)G {1,2,...,n}is asymptotically normal,and its order is loglog n Mehrdad and Zhu(2016)further studied the fine behaviors of the number of prime factors and obtained the large dev:iation and moderate deviation principles.Naturally,we can consider the similar problems of the polynomial ring.Liu(2004)proved the Erdos-Kac theorem on the polynomial ring.On the basis of the above works,we prove the large deviation principle for the number of polynomial prime factors on the finite field,which extends the results of Mehrdad and Zhu(2016)to the polynomial ring.