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Landau-Ramanujan Constant

المؤلف:  Berndt, B. C

المصدر:  Ramanujan,s Notebooks, Part IV. New York: Springer-Verlag

الجزء والصفحة:  ...

27-12-2020

1549

Landau-Ramanujan Constant

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

			(1)
			(2)
			(3)
			(4)
			(5)

(OEIS A001481), so , , , , , and so on. Then

(6)

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

(7)

where  and  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 ,  iff  is not divisible by a prime power  with  and  odd.

The constant has numerical value

(8)

(OEIS A064533). However, the convergence to the constant , known as the Landau-Ramanujan constant and sometimes also denoted , 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  is (OEIS A164775).

	7	1.062199
	43	0.922765
	330	0.867326
	2749	0.834281
	24028	0.815287
	216341	0.804123
	1985459	0.797109
	18457847	0.792198
	173229058	0.788587
	1637624156	0.785818

An exact formula for the constant is given by

(9)

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

(10)

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

(11)

where  is the Dirichlet beta function.

Another closed form is

(12)

where  is the Kronecker delta and  is the sum of squares Function.

W. Gosper used the related formula

(13)

where

(14)

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

Landau also proved the even stronger fact

(15)

where

			(16)
			(17)

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

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

(18)

has an asymptotic series

(19)

where  can be arbitrarily large and the  are constants with  (Moree and Cazaran 1999).

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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. 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: 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 ." 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." https://library.wolfram.com/infocenter/Demos/120/.