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Legendre's constant is a
mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
occurring in a formula constructed by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...
to approximate the behavior of the
prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ...
\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, (), as giving an approximation of y=\pi(x) with a "very satisfying precision". However, if one defines the real function B(x) by \pi(x)=\frac, and if B(x) converges to a real constant B as x tends to infinity, then this constant satisfies B = \lim_ \left( \log(x) - \right). Not only is it now known that the limit exists, but also that its value is equal to 1, somewhat less than Legendre's . Regardless of its exact value, the existence of the limit B implies the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
.
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...
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 In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
, under the precise form with an explicit estimate of the error term \pi(x) = \operatorname (x) + O \left(x e^\right) \quad\text x \to \infty (for some positive constant ''a'', where ''O''(...) is the
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
), 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 Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...
and La Vallée Poussin, Originally published in vol. 20 (1896)
Second scanned version
from a different library.
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 for values of x far beyond what was available to Legendre: Values up to \pi(10^) (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the Riemann R function.


References


External links

* {{Prime number conjectures Conjectures about prime numbers Mathematical constants 1 (number) Integers Analytic number theory