TOPICS
Search

Landau-Ramanujan Constant

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The Landau-Ramanujan constant is the constant K in the asymptotic formula for the count of sums of two squares. 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 such integers 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 or 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).

For a positive integer n, r_2(n)>0 iff every prime p with p=3 (mod 4) occurs with an even exponent in the prime factorization of n.

LandauRamanujanConstant

The constant has numerical value

K=0.764223653...
(8)

(OEIS A064533). However, the convergence to the constant K, sometimes also denoted lambda, is very slow. The following table summarizes the values of sqrt(lnx)S(x)/x 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 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 zeta(s) is the Riemann zeta function and 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.

An equivalent formula in terms of the polygamma function is

K=1/(sqrt(2))[1/(Psi(2)-1)]^(1/4)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_m is a Bernoulli number and psi_(m-1)(x) is the polygamma function of order m-1.

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 A227158), 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

Explore with Wolfram|Alpha

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. https://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, France: 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, Germany: 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, A164775, and A227158 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." https://library.wolfram.com/infocenter/Demos/120/.

Referenced on Wolfram|Alpha

Landau-Ramanujan Constant

Cite this as:

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

Subject classifications