The K-tuple Prime Number Theorem On Restricted Sets

The prime number theorem is a classical result in analytic number theory.Let π(x)be the number of primes not exceeding x,the prime number theorem is π(x)～x/log x →∞.The main aim of this thesis is to extend the prime number theorem to restricted subsets of primes.Suppose Ai(1≤i≤k)be the subset of primes.We defineπAi(x):=∑ p∈P p∈ Ai 1,and suppose that for each i,πAi(x)satisfiesπAi(x)=ρi x/logx+O(x/(logx)1+ε),where 0<ρi≤1,ε is a arbitrary positive number.Define A[k]=A1×A2×…×Ak,where Ai(?)P.Let P be the set of all prime numbers.We define Prabhu obtained the main terms of πA[k]and τA[k]for Ai={p≤x:p≡ai(mod q)}(1≤i≤k).The main results are the following asymptotic formulas and for any k≥2,where g(x)is a function satisfying g(x)→∞ and g(x)=o(loglogx)as x→∞.As consequences,we improve Prabhu’s results and obtain an explicit error term.We obtain the distributions on Chebotarev sets and the subset of primes represented by primitive positive definite binary quadratic form.