The K-tuple Prime Number Theorem On Restricted Sets

The prime number theorem is a classical result in analytic number theory.Let ?(x)be the number of primes not exceeding x,the prime number theorem is ?(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 define?Ai(x):=? p?P p? Ai 1,and suppose that for each i,?Ai(x)satisfies?Ai(x)=?i x/logx+O(x/(logx)1+?),where 0<?i?1,? 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 ?A[k]and ?A[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.