Legendre's constant

Constant of proportionality of prime number density
This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, commonly notated as ln(x) or loge(x).
The first 100,000 elements of the sequence an = log(n) − n/π(n) (red line) appear to converge to a value around 1.08366 (blue line).
Later elements up to 10,000,000 of the same sequence an = log(n) − n/π(n) (red line) appear to be consistently less than 1.08366 (blue line).

Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function π ( x ) {\displaystyle \pi (x)} . The value that corresponds precisely to its asymptotic behavior is now known to be 1.

Examination of available numerical data for known values of π ( x ) {\displaystyle \pi (x)} led Legendre to an approximating formula.

Legendre constructed in 1808 the formula

π ( x ) ≈ x log ⁡ ( x ) − B , {\displaystyle \pi (x)\approx {\frac {x}{\log(x)-B}},}

where B = 1.08366 {\displaystyle B=1.08366} (OEIS: A228211), as giving an approximation of π ( x ) {\displaystyle \pi (x)} with a "very satisfying precision".[1][2]

Today, one defines the value of B {\displaystyle B} such that

π ( x ) ∼ x log ⁡ ( x ) − B , {\displaystyle \pi (x)\sim {\frac {x}{\log(x)-B}},}

which is solved by putting

B = lim n → ∞ ( log ⁡ ( n ) − n π ( n ) ) , {\displaystyle B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),}

provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to 1 , {\displaystyle 1,} somewhat less than Legendre's 1.08366. {\displaystyle 1.08366.} Regardless of its exact value, the existence of the limit B {\displaystyle B} implies the prime number theorem.

Pafnuty Chebyshev proved in 1849[3] that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.[4]

It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term

π ( x ) = Li ⁡ ( x ) + O ( x e − a log ⁡ x ) as  x → ∞ {\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }

(for some positive constant a, where O(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin,[5] that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard[6] and La Vallée Poussin,[7] but without any estimate of the involved error term).

Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.

References

  1. ^ Legendre, A.-M. (1808). Essai sur la théorie des nombres. Courcier. p. 394.
  2. ^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 188. ISBN 0-387-20169-6.
  3. ^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
  4. ^ Pintz, Janos (1980). "On Legendre's Prime Number Formula". The American Mathematical Monthly. 87 (9): 733–735. doi:10.2307/2321863. ISSN 0002-9890. JSTOR 2321863.
  5. ^ La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
  6. ^ Sur la distribution des zéros de la fonction ζ ( s ) {\displaystyle \zeta (s)} et ses conséquences arithmétiques, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online Archived 2012-07-17 at the Wayback Machine
  7. ^ « Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361

External links

  • Weisstein, Eric W. "Legendre's constant". MathWorld.
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Prime number conjectures