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%I #18 Dec 12 2017 03:45:06
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%H J. Barkley Rosser, Lowell Schoenfeld, <a href="https://projecteuclid.org/euclid.ijm/1255631807"> Approximate formulas for some functions of prime numbers</a>, Illinois J. Math. 6 1962 64-94.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>.
%F C = 30*log(113)/113 = 1.255058712932479796968707476181244691689202758...
%e The maximum value for PrimePi(n)*log(n)/n occurs at n = 113.
%t $MaxPiecewiseCases=10000; sol=Maximize[{PrimePi[n]Log[n]/n, 1<n<10000}, n]; {N[sol[[1]], 100], sol[[2]]}
%Y Cf. A000720, A057835.
%K nonn,cons
%O 1,2
%A _Frank M Jackson_, Mar 14 2012
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