On the Asymptotic Behaviour of Sums of Two Squares : Theoretical and Numerical Studies of the Counting Function B(x)

Abstract: Studying the distributions of special numbers is a fundamental area of research in number theory. This paper studies the distribution of sums of two squares using analytic number theory. The asymptotic behaviour of the corresponding counting function, B(x), is the object of study. The rate of convergence for the relative error has not been in focus previously and the current formulas underperform when used to evaluate B(x). We introduce a Dirichlet series and use Perron’s formula to retrieve the first terms in the series. The integral is however unwieldy and numerically hard to calculate. Therefore the Dirichlet series is analytically extended and a branchcut is then introduced. Neglecting some parts of a new encircling contour in the complex plane allows for a simpler integral to arise. This can in turn be approximated into an explicit series formula with arbitrary many coefficients. The rest of the paper discusses a numerical method for evaluating the coefficients in the series and the results for the new formula. This paper is a small continuation of the work done by Daniel Shanks who calculated the first 2 coefficients with good precision. By calculating higher order coefficients we achieve even greater precision for large x. There also exists work that can be done to extend this report. Further extending analyticity of the Dirichlet series would allow for correction terms, hopefully reducing the asymptotic error to be close to the fourth root of x instead of the current square root error when assuming RH. Also the exact number of correct significant decimals in the calculated coefficients can be discussed further