Legendre's constant

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

log e (x)

.

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 $\pi (x)$ . The value that corresponds precisely to its asymptotic behavior is now known to be 1.

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

Legendre proposed in 1808 the formula $y={\frac {x}{\log(x)-1.08366}},$ (OEIS: A228211), as giving an approximation of $y=\pi (x)$ with a "very satisfying precision".^[1]^[2]

Today, one defines the real constant $B$ by $\pi (x)\sim {\frac {x}{\log(x)-B}},$ which is solved by putting $B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),$ 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 $B$ 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

$\pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty$

(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]^{: 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

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

Legendre's constant asymptotically approaching 1 for large values of $x$
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 $\pi (10^{29})$ (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the Riemann R function.

References

^ Legendre, A.-M. (1808). Essai sur la théorie des nombres [Essay on number theory] (in French). Courcier. p. 394.
^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 163. ISBN 0-387-20169-6.
^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
^ 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.
^ La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
^ Hadamard, Jacques (1896). "Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques" [On the distribution of the zeros of the function $\zeta (s)$ and its arithmetic consequences]. Bulletin de la Société Mathématique de France (in French). 24: 199–220. doi:10.24033/bsmf.545.
^ 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

Weisstein, Eric W. "Legendre's constant". MathWorld.

[1] Legendre, A.-M. (1808). Essai sur la théorie des nombres [Essay on number theory] (in French). Courcier. p. 394.

[P.Ribenboim-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 $\zeta (s)$ et ses conséquences arithmétiques" [On the distribution of the zeros of the function $\zeta (s)$ and its arithmetic consequences]. Bulletin de la Société Mathématique de France (in French). 24: 199–220. doi:10.24033/bsmf.545.

[7] 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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