A228211 Decimal expansion of Legendre's constant (incorrect, the true value is 1, as in A000007).

1, 0, 8, 3, 6, 6

Offset: 1

Author

Alonso del Arte, Nov 02 2013

Keywords

• nonn
• cons
• fini
• full

Comments

Included in accordance with the OEIS policy of listing incorrect but published sequences. The correct value of this constant is 1, by the prime number theorem pi(x) ~ li(x) = x/(log(x) - 1 - 1/log(x) + O(1/log^2(x))), where li is the logarithmic integral.

Before the prime number theorem was proved, it was believed that there was a constant A not equal to 1 that needed to be inserted in the formula pi(n) ~ n/(log(n) - A) to make it more precise. This number was Adrien-Marie Legendre's guess.

Panaitopol proved that x/(log(x) - A), where A is this constant, is an upper bound for pi(x) when x > 10^6. - John W. Nicholson, Feb 26 2018

Examples

			A = 1.08366.
		

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 80.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 163.
• Hans Riesel, Prime Numbers and Computer Methods for Factorization. New York: Springer (1994): 41 - 43.

Links

• Kevin Brown, Legendre's Prime Number Conjecture.
• Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016-2017.
• Laurenţiu Panaitopol, Several Approximations of pi(x), Math. Ineq. Appl. 2(1999), 317-324.
• Eric Weisstein's World of Mathematics, Legendre's Constant.

Crossrefs

Cf. A000007.

Formula

Believed at one time to be lim_{n -> oo} A(n) in pi(n) = n/(log(n) - A(n)).

Extensions

Edited by N. J. A. Sloane, Nov 13 2014