Legendre's constant is the number 1.08366 in Legendre's guess at the prime
number theorem
with .
Legendre first published a guess the form
in his Essai sur la Théorie des Nombres (Edwards 2001, p. 3; Havil 2003, p. 177), but in the third edition (renamed Théorie des nombres),
modified it to the form above (Derbyshire 2004, pp. 55 and 369).
This expression is correct to leading term only, since it is actually true that this limit approaches 1 (Rosser and Schoenfeld 1962, Panaitopol 1999).
Abel, N. H. Letter to Holmboe. 4 August 1823. In Niels Henrik Abel, Memorial, Publié à l'Occasion du Centenaire de sa Naissance.Chebyshev,
P. L. "Sur la fonction qui détermine la totalité des nombres
premiers inferieurs à une limite donnée." J. math. pures appl.17,
1852.Derbyshire, J. Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Penguin, 2004.Dirichlet, P. G. L. "Sur l'usage
des series infinies dans la théorie des nombres." J. reine angew.
Math.18, 257-274, 1838. Reprinted in Werke, Vol. 1, pp. 359-374.Edwards,
H. M. Riemann's
Zeta Function. New York: Dover, pp. 3-4, 2001.Havil, J.
Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Le
Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 147, 1983.Legendre,
A. M. Essai sur la Théorie des Nombres. Paris: Duprat, 1808.Panaitopol,
L. "Several Approximations of ." Math. Ineq. Appl.2, 317-324, 1999.Ribenboim,
P. The
New Book of Prime Number Records. New York: Springer-Verlag, 1996.Rosser,
J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime
Numbers." Ill. J. Math.6, 64-94, 1962.Wagon, S.
Mathematica
in Action. New York: W. H. Freeman, pp. 28-29, 1991.
TOPICS
.
Legendre first published a guess the form
." Math. Ineq. Appl. 2, 317-324, 1999.Ribenboim,
P. The
New Book of Prime Number Records. New York: Springer-Verlag, 1996.Rosser,
J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime
Numbers." Ill. J. Math. 6, 64-94, 1962.Wagon, S.
Mathematica
in Action. New York: W. H. Freeman, pp. 28-29, 1991.