The Number Of Square-free Integers As Sums Of Two Squares

In this paper, we mainly study the arithmetic function r(n). It denotes the number of representations of the integer n as a sum of two squares. For this arithmetic function Gauss studied and firstly proved that The exponent 1/2 was improved to θ<1/3 later.In 1979, K. H. Fischer [7] studied square-free numbers as sums of two squares, and proved that μ(n) denotes the Mobius function, whereIn 1982, E. Kratzel [15] studied the short interval case, and proved that Q(x+y)-Q(x)=Ay+o(y), whereIn 2006, Zhai [26] improved the error term of E. Kratzel’s [15] result, and proved that if the estimate P(x)=O(xθ) holds, then Q(x+h)-Q(x)= Ah+O(hx-ε/2+xθ+ε).where A is a constant,1/4<θ<1/3, h= o(x). Particularly, the above asymptotic formula is true for θ=131/416. In this paper we will mainly study the asymptotic formula ofIn this paper we will mainly use the method about dealing with convo-lution in A. Ivic [12] and the conclusion in J. B. Friedlander, H. Iwaniec [8] theorem 4.2 and the theorem in page 226 of M. Kiihleitner, W. G. Nowak [17].In this paper we will get the following two results:There are three sections in this paper. In the first section, we mainly introduce the result of the former researcher. In the second section, we will introduce many definitions and basic knowledge which will be used in this paper. Lastly, we will introduce some lemmas and prove the main theorems.