Multiple Mertens' Second Theorem In The Polynomial Ring Over A Finite Field

In 1874,Mertens obtained the following estimation which is now known as the Mertens' second theorem:(?)where B(?0.261)is the Mertens' constant.It has been appeared many times in the textbooks on number theory,and it serves as an important foundation for investigating many problems in number theory.In the early days,it was known that the integer ring Z shares many arithmetic prop-erties with the polynomial ring over a finite field Fq[T].In 1979,Knopfmacher proved an analogue of Mertens' second theorem in Fq[T].In fact,he estimated the sum(?),where P is the set of monic irreducible polynomials over IFq[T].In this thesis,we provide an elementary proof for Mertens' second theorem over IFq[T]by using Abel summation formula and the prime number theorem in Fq[T].In addition,we also generalize this theorem to multiple cases in IFq[T]by using the Dirichlet's hyperbola method.That is,we shall estimate the sum(?)where k is any positive integer.