Mean-square Value Of The Error Term On Three-dimensional Divisor Problem Of (a, A, B) Type

To study the mean value of divisor problem d(n)=Σ_d|n1 is one of the famous number theory problems. LetD(x) =(?)d(n),to estimate the better upper bound of the error term is called Dirichlet divisor problem. In 1849, Dirichlet first proved that the following asymptotic formulaholding, withâ–³(x)=O(x^1/2). The exponent 1/2 was improved by many authors. It is conjectured that for allε> 0,which is supported by the classical mean-square resultNow we suppose 1≤a≤b≤c be real numbers, and defineTo study the general estimates of D(a, b, c: x) for any combination (a, b, c) is the well-known three-dimensional divisor problem. For a special case of threedimensional divisor problem, we writeD(a, a, b; x) = H(a, a, b; x) +â–³(a, a, b; x), a < bwhereUsing analytic methods, we can easily get the main term of the above problem. Thus we only need to seek better upper bound for the error termâ–³(a, a, b; x). We often change this problem into studying the mean-square value of the error term. In the first chapter of this paper, we will study the mean value of the error term on the three-dimensional divisor problem of (a, a, b) type, namely, the mean value ofâ–³(a,a,b). Noticing that the case a=1 has already been resolved, and, if a|b or (a, b) = d＞1, they can also be reduced to the cases that have been resolved. In this paper, we are interested in the case a≥2 and (a, b) = 1, and we've got better results for the case b＞5a/2, especially. We will proveTheorem Suppose T≥2, a, b be integers with a≥2及(a, b) = 1, then wehavewithIn the second chapter of this paper, let r(n) denote the number of representations of the integer n as a sum of two squares, q3(n) denote the characteristic function of the set of cube-free integers, and P(x) the error term of the Gauss circle problem. Letit is easy to check that qa(n)r(n) denotes the number of representations of a cubefree number by two squares. We will study the short interval case and prove that if the estimate P(x) = O(x^θ) holds, then for x^θ+ε≤y≤x, we haveQ₃(x + y) - Q₃(x) = C_y + O(yx^-ε／2+x^θ+εwhere C is a constant. In particular this asymptotic formula is true forθ=131/416.