A Special Grid

in this dissertation, which contains two chapters, mainly by virtue of vari- ous techniques for bounding exponential sums, we give some better estiniates of the error terms that belong to a. kind of special problems of lattice points under the assumption of Riemann Hypothesis (RH). In chapter 1, using the well-known method we firstly obtain the asymptot- ic formula of ^*(a, b; x), which is the error term of two-dimensional co-prime lattice points problem D*(a, b; x). Then resorting to the symmetric expression of ^{a, h; x} that is the error term of the general two-dimensional divisor prob- lein D(a, b; x), we can convert the upper bound estimate of ^*(a, b; 4 into the estimate of the sum ji(m) Z ^k(xrnc¹n²). In the end, exploiting van n n拁N der corput method, Heath-Brown method and some results given by the large Sieve method and coml)ining with Heath-Brown identity, we propose a sharper estiiiiate of b; 4, which iml)roVeS the weaker results given l)y lvic and Nowark. Similarly, in chapter 2, employing the symmetric and non-symmetric ex- pressions of the error term ^{a, a, b; 4 that belong to the three-dimensional divisor problem of (a, a, h) ty}e D(a, a, b; 4, we will find that the estimate of the sums a(x) 3 3 b(wx1y2Z3) and 3 ,u(d) 3 wd1m2 can y?Y z-.Z dD niM lead to the upper bound ecltimate of the error term a, 4 of the three- dirnensiona.l co-prime lattice points problem D*(a, a, b; 4. Due to the better estimates of the former sums, we give a sharper estimate of the error term a, b; r). From the proof of the dissertation, it is obvious that our results benefit from these ideas: (1) employing heath-Brown identity, we can give better results to some exponential sums containing the special number theory function m; (2) by converting the estimate of some special high-dimentional exponential sums into the estimate of ]ower-climentionai ones, we arrive at some better rcs u it s.</sup>