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%I #8 May 14 2012 12:54:14
%S 1,3,5,0,2,2,3,3,6,8,7,3,2,2,5,8,2,1,1,7,0,5,7,5,4,9,6,4,8,3,8,1,2,4,
%T 7,1,0,3,6,0,4,2,6,1,3,8,8,9,3,5,3,6,3,3,4,8,4,9,3,7,2,7,5,7,0,9,9,5,
%U 4,5,2,1,0,8,8,9,1,9,0,9,2,0,5,0,5,7,2,2,2,2,3,5,0,9,9,5,1,6,7,2
%N Decimal expansion of constant C = maximum value that sigma(n)*log(n^2)/n^2 reaches where sigma(n) = (sum of primes <= n), A034387.
%C From the prime number theorem it can be shown that the Prime sums function sigma(n) = (sum of primes <= n) ~ n^2/log(n^2). Consequently, the function sigma(n)*log(n^2)/n^2 tends to 1 as n tends to infinity, however it has a maximum value of 1.3502233687.... when n=7. In precise terms this constant is 34*log(7)/49 and it provides an upper bound for sigma(n), i.e. sigma(n) <= (34*log(7)/49)*n^2/log(n^2) for all n > 1.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>. Illinois J. Math. 6 (1962), pp. 64-94.
%F The maximum value for sigma(n)*log(n^2)/n^2 occurs at n = 7, so C = 34*log(7)/49.
%e 1.350223368732258211705754964838124710360426138...
%t table=Table[Sum[Prime[k], {k, 1, PrimePi[n]}]/(n^2/(2 Log[n])), {n, 2, 10^4}]; max=Max[table]; n=1; While[table[[n]]!=max, n++]; Print[N[max, 100], " at n = ", n+1]
%o (PARI) log(7)*34/49 \\ _Charles R Greathouse IV_, May 14 2012
%Y Cf. A034387.
%K nonn,cons
%O 1,2
%A _Frank M Jackson_, May 14 2012
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