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Legendre's constant

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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, also commonly written 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 proposed in 1808 the formula y = x log ( x ) 1.08366 , {\displaystyle y={\frac {x}{\log(x)-1.08366}},} (OEISA228211), as giving an approximation of y = π ( x ) {\displaystyle y=\pi (x)} with a "very satisfying precision".

Today, one defines the real constant B {\displaystyle B} by 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, somewhat less than Legendre's 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 that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.

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, that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard and La Vallée Poussin, 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.

Numerical values

Using known values for π ( x ) {\displaystyle \pi (x)} , we can compute B ( x ) = log x x π ( x ) {\displaystyle B(x)=\log x-{\frac {x}{\pi (x)}}} for values of x {\displaystyle x} far beyond what was available to Legendre:

Legendre's constant asymptotically approaching 1 for large values of x {\displaystyle x}
x B(x) x B(x) x B(x) x B(x)
10 0.605170 10 1.029660 10 1.015148 10 1.010176
10 0.955374 10 1.027758 10 1.014637 10 1.009943
10 1.073644 10 1.026085 10 1.014159 10 1.009720
10 1.087571 10 1.024603 10 1.013712 10 1.009507
10 1.076332 10 1.023281 10 1.013292 10 1.009304
10 1.070976 10 1.022094 10 1.012897 10 1.009108
10 1.063954 10 1.021022 10 1.012525 10 1.008921
10 1.056629 10 1.020050 10 1.012173 10 1.008742
10 1.050365 10 1.019164 10 1.011841 10 1.008569
10 1.045126 10 1.018353 10 1.011527 10 1.008403
10 1.040872 10 1.017607 10 1.011229 10 1.008244
10 1.037345 10 1.016921 10 1.010946 10 1.008090
10 1.034376 10 1.016285 10 1.010676 10 1.007942
10 1.031844 10 1.015696 10 1.010420 10 1.007799

Values up to π ( 10 29 ) {\displaystyle \pi (10^{29})} (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the Riemann R function.

References

  1. Legendre, A.-M. (1808). Essai sur la théorie des nombres [Essay on number theory] (in French). Courcier. p. 394.
  2. Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 163. 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. Hadamard, Jacques (1896). "Sur la distribution des zéros de la fonction ζ ( s ) {\displaystyle \zeta (s)} et ses conséquences arithmétiques" [On the distribution of the zeros of the function ζ ( s ) {\displaystyle \zeta (s)} and its arithmetic consequences]. Bulletin de la Société Mathématique de France (in French). 24: 199–220. doi:10.24033/bsmf.545.
  7. de la Vallée Poussin, Charles Jean (1897). Recherches analytiques sur la théorie des nombres premiers [Analytical research on prime number theory] (in French). Brussels: Hayez. pp. 183–256, 281–361. Originally published in Annales de la société scientifique de Bruxelles vol. 20 (1896). Second scanned version, from a different library.

External links

Prime number conjectures
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