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Landau-Ramanujan Constant

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Let S(x) denote the number of positive integers not exceeding x which can be expressed as a sum of two squares (i.e., those n<=x such that the sum of squares function r_2(n)>0). For example, the first few positive integers that can be expressed as a sum of squares are

1=0^2+1^2
(1)
2=1^2+1^2
(2)
4=0^2+2^2
(3)
5=1^2+2^2
(4)
8=2^2+2^2
(5)

(OEIS A001481), so S(1)=1, S(2)=2, S(4)=3, S(5)=4, S(8)=5, and so on. Then

lim_(x->infty)(sqrt(lnx))/xS(x)=K,
(6)

as proved by Landau (1908), where K is a constant. Ramanujan independently stated the theorem in the slightly different form that the number of numbers between A and x which are either squares of sums of two squares is

S(x)=Kint_A^x(dt)/(sqrt(lnt))+theta(x),
(7)

where K approx 0.764 and theta(x) is very small compared with the previous integral (Berndt and Rankin 1995, p. 24; Hardy 1999, p. 8; Moree and Cazaran 1999).

Note that for n>1, r_2(n)>0 iff n is not divisible by a prime power p^m with p=3 (mod 4) and m odd.

LandauRamanujanConstant

The constant has numerical value

K=0.764223653...
(8)

(OEIS A064533). However, the convergence to the constant K, known as the Landau-Ramanujan constant and sometimes also denoted lambda, is very slow. The following table summarizes the values of the left side of equation (7) for the first few powers of 10, where the sequence of S(10^n) is (OEIS A164775).

xS(x)(sqrt(lnx))/xS(x)
10^171.062199
10^2430.922765
10^33300.867326
10^427490.834281
10^5240280.815287
10^62163410.804123
10^719854590.797109
10^8184578470.792198
10^91732290580.788587
10^(10)16376241560.785818

An exact formula for the constant is given by

K=1/(sqrt(2))product_(p prime ; = 3 (mod 4))(1-1/(p^2))^(-1/2)
(9)

(Landau 1908; Le Lionnais 1983, p. 31; Berndt 1994; Hardy 1999; Moree and Cazaran 1999), and an equivalent formula is given by

K=pi/4product_(p prime ; = 1 (mod 4))(1-1/(p^2))^(1/2).
(10)

Flajolet and Vardi (1996) give a beautiful formula with fast convergence

K=1/(sqrt(2))product_(n=1)^infty[(1-1/(2^(2^n)))(zeta(2^n))/(beta(2^n))]^(1/2^(n+1)),
(11)

where beta(s) is the Dirichlet beta function.

Another closed form is

K=lim_(n->infty)(sqrt(lnn))/nsum_(k=1)^n[1-delta_(0,r_2(k))],
(12)

where delta_(i,j) is the Kronecker delta and r_2(k) is the sum of squares Function.

W. Gosper used the related formula

K=1/2[1/(Psi(2)-1)]^(sqrt(2))product_(k=2)^infty[1/(-Psi(2^k)-1)]^(1/(2^(k+1))),
(13)

where

Psi(m)=(mpsi_(m-1)(1/4))/(pi^m(2^m-1)4^(m-1)B_m),
(14)

where B_n is a Bernoulli number and psi(x) is a polygamma function (Finch 2003).

Landau also proved the even stronger fact

lim_(x->infty)((lnx)^(3/2))/(Kx)[S(x)-(Kx)/(sqrt(lnx))]=C,
(15)

where

C=1/2[1-ln((pie^gamma)/(2L))]-1/4d/(ds)[ln(product_(p prime; = 3 (mod 4))1/(1-p^(-2s)))]_(s=1)
(16)
=0.581948659...
(17)

(OEIS A085990), e is the base of the natural logarithm, gamma is the Euler-Mascheroni constant, and L is the lemniscate constant.

Landau's method of proof can be extended to show that

S(x)∼Kx/(sqrt(lnx))
(18)

has an asymptotic series

S(x)=Kx/(sqrt(lnx))[1+(c_1)/(lnx)+(c_2)/((lnx)^2)+...+(c_n)/((lnx)^n)+O(1/((lnx)^(n+1)))],
(19)

where n can be arbitrarily large and the c_j are constants with c_1=C (Moree and Cazaran 1999).

See also

Landau Constant, Landau-Kolmogorov Constants, Square Number

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References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 60-66, 1994.Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., pp. 25, 47, and 49, 1995.Finch, S. R. "Landau-Ramanujan Constant." §2.3 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 98-104, 2003.Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 9-10, 55, and 60-64, 1999.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.Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, Bd. II, 2nd ed. New York: Chelsea, pp. 641-669, 1953.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Moree, P. and Cazaran, J. "On a Claim of Ramanujan in His First Letter to Hardy." Expos. Math. 17, 289-312, 1999.Selberg, A. Collected Papers, Vol. 2. Berlin: Springer-Verlag, pp. 183-185, 1991.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.Sloane, N. J. A. Sequences A001481/M0968, A064533, A085990, and A164775 in "The On-Line Encyclopedia of Integer Sequences."Stanley, G. K. "Two Assertions Made by Ramanujan." J. London Math. Soc. 3, 232-237, 1928.Stanley, G. K. Corrigendum to "Two Assertions Made by Ramanujan." J. London Math. Soc. 4, 32, 1929. Wolfram Research, Inc. "Computing the Landau-Ramanujan Constant." http://library.wolfram.com/infocenter/Demos/120/.

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Landau-Ramanujan Constant

Cite this as:

Weisstein, Eric W. "Landau-Ramanujan Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Landau-RamanujanConstant.html

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