## Landau-Ramanujan Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let denote the number of Positive Integers not exceeding which can be expressed as a sum of two squares, then

 (1)

as proved by Landau (1908) and stated by Ramanujan. The value of (also sometimes called ) is
 (2)

(Hardy 1940, Berndt 1994). Ramanujan found the approximate value . Flajolet and Vardi (1996) give a beautiful Formula with fast convergence
 (3)

where
 (4)

is the Dirichlet Beta Function, and is the Hurwitz Zeta Function. Landau proved the even stronger fact
 (5)

where
 (6)

Here,
 (7)

is the Arc Length of a Lemniscate with (the Lemniscate Constant to within a factor of 2 or 4), and is the Euler-Mascheroni Constant.

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 60-66, 1994.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lr/lr.html

Flajolet, P. and Vardi, I. Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 61-63, 1940.

Landau, E. Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate.'' Arch. Math. Phys. 13, 305-312, 1908.

Shanks, D. The Second-Order Term in the Asymptotic Expansion of .'' Math. Comput. 18, 75-86, 1964.

Shanks, D. Non-Hypotenuse Numbers.'' Fibonacci Quart. 13, 319-321, 1975.

Shanks, D. and Schmid, L. P. Variations on a Theorem of Landau. I.'' Math. Comput. 20, 551-569, 1966.

Shiu, P. Counting Sums of Two Squares: The Meissel-Lehmer Method.'' Math. Comput. 47, 351-360, 1986.