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Legendre's constant

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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 . The value that corresponds precisely to its asymptotic behavior is now known to be 1.

Examination of available numerical data for known values of led Legendre to an approximating formula.

Legendre proposed in 1808 the formula (OEISA228211), as giving an approximation of with a "very satisfying precision".[1][2]

Today, one defines the real constant by which is solved by putting 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 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

(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]: 183–256, 281–361 [page needed] 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

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Using known values for , we can compute for values of far beyond what was available to Legendre:

Legendre's constant asymptotically approaching 1 for large values of
x B(x) x B(x) x B(x) x B(x)
102 0.605170 1016 1.029660 1030 1.015148 1044 1.010176
103 0.955374 1017 1.027758 1031 1.014637 1045 1.009943
104 1.073644 1018 1.026085 1032 1.014159 1046 1.009720
105 1.087571 1019 1.024603 1033 1.013712 1047 1.009507
106 1.076332 1020 1.023281 1034 1.013292 1048 1.009304
107 1.070976 1021 1.022094 1035 1.012897 1049 1.009108
108 1.063954 1022 1.021022 1036 1.012525 1050 1.008921
109 1.056629 1023 1.020050 1037 1.012173 1051 1.008742
1010 1.050365 1024 1.019164 1038 1.011841 1052 1.008569
1011 1.045126 1025 1.018353 1039 1.011527 1053 1.008403
1012 1.040872 1026 1.017607 1040 1.011229 1054 1.008244
1013 1.037345 1027 1.016921 1041 1.010946 1055 1.008090
1014 1.034376 1028 1.016285 1042 1.010676 1056 1.007942
1015 1.031844 1029 1.015696 1043 1.010420 1057 1.007799

Values up to (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the Riemann R function.

References

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  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 et ses conséquences arithmétiques" [On the distribution of the zeros of the function and its arithmetic consequences]. Bulletin de la Société Mathématique de France (in French). 24: 199–220. doi:10.24033/bsmf.545. Archived from the original on 2024-04-17.
  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.
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