The Generation Of Dirichlet Divisor Problem Over Pjatecki(?)-(?)apiro Prime Theorem

In this dissertation, we study the generation of Dirichlet divisor problem on the condition of Pjateckii-Sapiro prime theorem.In number theory, there is a famous problem which is about the estimation of A(x), the error term in the asymptotic formula for (?) d(n), where d(n) is thenumber of ways n can be written as a product of two factors. Up to now, peoplehave got many goodresults,Δ(x)<< x^1/2Dirichlet(1849)Δ(x)<1/3 log xVoronoi(1904)Δ(x)<27/82van der Corput(1928)Δ(x)<346/1067Kolesnik(1973)Δ(x)<35/108 log² xKolesnik(1982)Δ(x)<23/73 log ^{315/146 x}Huxley(1993)Δ(x) <131/416 log^26957/8320 x Huxley(2003)but there is some distance to our expected goal.Many people believe that for any small positive real number ε, we haveΔ(x)<1/4+ε.Similarly, people have considered the divisor problems on certain conditions. Health-Brown, Iwanicc stdudied the problem under the condition of arithmetic progression.Let c be a positive constant and let Ï€_c(x) denote the number of n ≤ x for which [n^c] is prime. Here [9] denotes, as usual, the integral part of θ. It is easy toprove, using the prime number theorem, thatvrc(x) a;/(cloga;)for 0 < c < 1. When 1 < c < 2, one still expects the above formula to hold, but if c = 2, then [nc] is always a aquare, so that ^{x) — 0. Pjateckii-SapW11! showed it does indeed hold for 1 < c < |y and Heath-Brown^ extended this range to g§ = 1.1404While the problem of representing primes by linear polynomials is completely solved by Dirichlet's Theorem on primes in arithmetic progressions, it is not known if there exists any quadratic polynomial that takes infinitely many prime values. One can therefore look on the investigation of the above formula for 1 < c < || as an intermediate problem for our progress towards the quadratic case. It is worth remarking that Deshouillers f13' has shown that nc(x) tends to infinity for almost all positive real c (in the sense of Lebesgue measure);however, this result provides no specific value of c.In Chapter 1, Our dissertation study the divisor problem under the condition of PjateckiT-Sapiro prime theorem, which gives us the asymptotic formula for ^ 1>where x > l,p is a prime and c is some constant such that 0 < c < 1, and we get Theorem 1.1 Let 0 < c < |, thenwhere d > 0 is some constant.In Chapter 2, we study the divisor problem over the set of k-free numbers. When k > 4, as it is hard to get a better estimation, we consider the problem in a short interval. We getTheorem 2.1^2d(n)fk(n) = Axlogx + Bx + Ek(x),n