Studies On Several Analytic Problems In Number Theory

In this thesis,we study the following four analytic problems in number theory.1)Ramanujan expansion in a polynomial ring of one variable over a finite field Ramanujan sums are introduced by Ramanujan around one hundred yours ago.Combing Delange,Ushiroya and Toth’s works from 1976 to 2017,we have that the multi-variable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums,this is an analogue of the Fourier expansion for periodic functions in the classical analysis.In this part,based on the predecessors’ works,we further investigate the properties of Ramanujan sums in the polynomial ring Fq[T],and show that the multi-variable arithmetic functions defined on Fq[T]can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.2)Sums of products of polynomial Ramanujan sums In this part,by further studying the orthogonal properties of polynomial Ramanujan sums,we establish an iden-tity between polynomial Ramanujan sums and the number of solutions of the system of polynomial congruence equations over the ring IFq[T].3)Multiple Mertens’ theorem In 1874,Mertens got two asymptotic formulas about the reciprocal sum of prime numbers,which are named Mertens’first and second theorem,respectively.These two theorems have been appeared in many textbooks on number theory.In this part,we generalize them to get explicit remained formulas for products of arbitrary number of primes by the inductive method,and the main term in our formula for Mertens’ second theorem is expressed by a recurrence formula for the special values of Riemann zeta functions.4)The Menon-Sury identities in a polynomial ring of one variable over a finite field In 2009,Sury got the following classical Menon-Sury’s identity(?)where n is a positive integer,Zn*is the unit group of ring Zn=Z/nZ,gcd(,)is the greatest common divisor,φ is the Euler φ function,and σn(n)=∑d/n dr.In this part,we prove the Menon-Sury type identity with several Dirichlet characters and several addition characters for the general arithmetic function defined on Fq[T].From the viewpoint of analysis,this give an explicit expression of Fourier transformation for general arithmetic functions on IFq[T].