Legendre's constant

Constant of proportionality of prime number density From Wikipedia, the free encyclopedia

Legendre's constant

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.

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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).
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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).

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

Legendre proposed in 1808 the formula (OEIS: A228211), 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

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

More information Legendre's constant asymptotically approaching 1 for large values of ...
Legendre's constant asymptotically approaching 1 for large values of
xB(x) xB(x) xB(x) xB(x)
1020.605170 10161.029660 10301.015148 10441.010176
1030.955374 10171.027758 10311.014637 10451.009943
1041.073644 10181.026085 10321.014159 10461.009720
1051.087571 10191.024603 10331.013712 10471.009507
1061.076332 10201.023281 10341.013292 10481.009304
1071.070976 10211.022094 10351.012897 10491.009108
1081.063954 10221.021022 10361.012525 10501.008921
1091.056629 10231.020050 10371.012173 10511.008742
10101.050365 10241.019164 10381.011841 10521.008569
10111.045126 10251.018353 10391.011527 10531.008403
10121.040872 10261.017607 10401.011229 10541.008244
10131.037345 10271.016921 10411.010946 10551.008090
10141.034376 10281.016285 10421.010676 10561.007942
10151.031844 10291.015696 10431.010420 10571.007799
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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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