| In this thesis,we study the distribution of the primes in special sequences and representation functions on abelian groups.1.Distribution of the primes in special sequences.The distribution of the primes of the forms[nα]and[nα+β]are studied extensively.We will consider several new type problems on the distribution of the primes involving the ceiling function.In this thesis,we introduce the following two functionsπ’θ(n)and π"θ(n).For any real number θ with 0<θ ≤1,let π’θ(n)be the number of integers k with 1 ≤k≤nθ such that[n/k]is prime and let π"θ(n)be the number of primes p for which there exists an integer k with 1 ≤k≤nθsuch that p=[n/k],where[x]denotes the least integer not less than x.These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial Bn(x)-Bn.We study asymptotic properties of π’θ(n)andπ"θ(n).In particular,we give the asymptotic formulae of π’θ(n)and π"θ(n)for 1/2<θ≤1 and the asymptotic formulae of mean values of π’θ(n)and π"θ(n)for 0<θ≤1/2.Our results have been published in Int.J.Number Theory.2.Representation functions on abelian groups.Let G denote a finite abelian group,A a nonempty subsets of G,and h≥2 an integer.For g ∈G,let RA,h(g)denote the number of solutions of the equation x1+…+xh=g with xi ∈A(1 ≤i≤h).In this thesis,We improve the related results of Kiss,Rozgonyi and Sandor and prove that RG\A,h(g)+(-1)hRA,h(g)does not depend on g.In particular,if h is even,then RA,h(g)＝RG\A,h(g)for some g∈G if and only if RA,h(g)=RG\A,h(g)for all g∈G if and only if |G|=2|A|.If h>1 is odd and RA,h(g)=RG\A,h(g)for all g∈G,then |A| is even.Our results have been published in Bull.Aust.Math.Soc.. |
