| 释义 |
Landau-Ramanujan ConstantN.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Let  denote the number of Positive Integers not exceeding  which can be expressed as a sum oftwo squares, then
 | (1) |
 (also sometimes called  ) is
 | (2) |
 . Flajolet and Vardi (1996) give a beautifulFormula with fast convergence
 | (3) |
 | (4) |
 is the Hurwitz Zeta Function. Landau proved the evenstronger fact
 | (5) |
Here,
 | (7) |
 (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 Unpublished manuscript. 1996.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.
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