The Average Estimate Of The Number Of Special Quadratic Factors

For a positive integern,let ??n?denote the ordinary divisor function.As we all know,??n?plays an important role in number theory.Main research of this paper is the mean value of divisor function of the binary quadratic form m²+n²,i.e.,the asymptotic formula of the sumWe mainly study from the following aspects.Firstly,we briefly introduce the back-ground,research status and the main result of the thesis.Next,we list some related concepts and lemmas,which provide a strong basis for the proof of the following theorem.At last,thanks to the average estimate of multiplicative function,circle method and the estimate of exponential sums and so on,we give the complete proof of the main theorem,then,we get the following result S?x?= A₁x²logx+A₂X²+O(x^3/2log¹¹x)holds for x?+?,where A₁ and A₂ are certain constants.The contribution of the thesis is to improve the error O(x^3/2+?)claimed by Yu[19]to O(x^3/2log¹¹x).