Landau-Ramanujan Constant

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

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

$$\lim_{x\to\infty} {\sqrt{\ln x}\over x} S(x)=K,$$ (1)

as proved by Landau (1908) and stated by Ramanujan. The value of $K$ (also sometimes called $\lambda$) is

$$K=\sqrt{{\textstyle{1\over 2}}\prod_{\scriptstyle p{\rm\ a\ ... ...v\ 3\ ({\rm mod\ } 4)} {1\over 1-p^{-2}}} = 0.764223653\ldots$$ (2)

(Hardy 1940, Berndt 1994). Ramanujan found the approximate value $K=0.764$. Flajolet and Vardi (1996) give a beautiful Formula with fast convergence

$$K={1\over\sqrt{2}}\prod_{n=1}^\infty\left[{\left({1-{1\over 2^{2^n}}}\right){\zeta(2^n)\over\beta(2^n)}}\right]^{1/(2^n+1)},$$ (3)

where

$$\beta(s)\equiv {1\over 4^s}[\zeta(s, {\textstyle{1\over 4}})-\zeta(s, {\textstyle{3\over 4}})]$$ (4)

is the Dirichlet Beta Function, and $\zeta(z,a)$ is the Hurwitz Zeta Function. Landau proved the even stronger fact

$$\lim_{x\to\infty} {(\ln x)^{3/2}\over Kx}\left[{S(x)-{Kx\over\sqrt{\ln x}}}\right]=C,$$ (5)

where

$$C \equiv {1\over 2}\left[{1-\ln\left({\pi e^\gamma\over L}\right)}\right]-... ...le p{\rm\ prime}\atop\scriptstyle p=4k+3}{1\over p^{-2s}}}\right)}\right]_{s=1}$$

$$= 0.581948659\ldots.$$ (6)

Here,

$$L=5.2441151086\ldots$$ (7)

is the Arc Length of a Lemniscate with $a=1$ (the Lemniscate Constant to within a factor of 2 or 4), and $\gamma$ 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 Unpublished manuscript. 1996.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 $B(x)$.'' 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.