Legendre's constant
the number 1, as occurring in a formula conjectured by Legendre
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}},}
(OEIS: A228211), as giving an approximation of
y
=
π
(
x
)
{\displaystyle y=\pi (x)}
with a "very satisfying precision".
However, if one defines the real function
B
(
x
)
{\displaystyle B(x)}
by
π
(
x
)
=
x
log
(
x
)
−
B
(
x
)
,
{\displaystyle \pi (x)={\frac {x}{\log(x)-B(x)}},}
and if
B
(
x
)
{\displaystyle B(x)}
converges to a real constant
B
{\displaystyle B}
as
x
{\displaystyle x}
tends to infinity, then this constant satisfies
B
=
lim
x
→
∞
(
log
(
x
)
−
x
π
(
x
)
)
.
{\displaystyle B=\lim _{x\to \infty }\left(\log(x)-{x \over \pi (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 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.
“Legendre's constant” enters the record as the number 1, as occurring in a formula conjectured by Legendre. Crown Archives preserves that source wording while asking what Legendre's, constant and number can confirm, complicate or overturn.
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