A209883 Decimal expansion of constant C = maximum value that PrimePi(n)*log(n)/n reaches where PrimePi(n) is the number of primes less than or equal to n, A000720.

1, 2, 5, 5, 0, 5, 8, 7, 1, 2, 9, 3, 2, 4, 7, 9, 7, 9, 6, 9, 6, 8, 7, 0, 7, 4, 7, 6, 1, 8, 1, 2, 4, 4, 6, 9, 1, 6, 8, 9, 2, 0, 2, 7, 5, 8, 0, 6, 2, 7, 4, 1, 7, 1, 5, 4, 1, 7, 7, 9, 1, 5, 1, 3, 8, 0, 8, 0, 2, 8, 4, 7, 0, 5, 0, 2, 4, 0, 2, 6, 7, 3, 6, 7, 3, 3, 2, 4, 8, 0, 5, 9, 7, 3, 4, 1, 7, 3, 6, 5, 8, 3

Offset: 1

Author

Frank M Jackson, Mar 14 2012

Keywords

• nonn
• cons

Comments

The prime number theorem states that PrimePi(n) ~ n/log(n). Consequently, the function PrimePi(n)*log(n)/n tends to 1 as n tends to infinity, however it has a maximum value of 1.2550587.... when n=113. In precise terms this constant is 30*log(113)/113 and it provides an upper bound for PrimePi(n), i.e. PrimePi(n) <= (30*log(113)/113)*n/log(n) for all n>1.

Examples

			The maximum value for PrimePi(n)*log(n)/n occurs at n = 113.
		

Links

• J. Barkley Rosser, Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94.
• Eric Weisstein's World of Mathematics, Prime Counting Function.

Crossrefs

Cf. A000720, A057835.

Programs

Mathematica

$MaxPiecewiseCases=10000; sol=Maximize[{PrimePi[n]Log[n]/n, 1
				

Formula

C = 30*log(113)/113 = 1.255058712932479796968707476181244691689202758...