A212394 Decimal expansion of constant C = maximum value that sigma(n)*log(n^2)/n^2 reaches where sigma(n) = (sum of primes <= n), A034387.

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, 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, 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

Offset: 1

Author

Frank M Jackson, May 14 2012

Keywords

• nonn
• cons

Comments

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.

Examples

			1.350223368732258211705754964838124710360426138...
		

Links

• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Crossrefs

Cf. A034387.

Programs

Mathematica

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]

PARI

log(7)*34/49 \\ Charles R Greathouse IV, May 14 2012

Formula

The maximum value for sigma(n)*log(n^2)/n^2 occurs at n = 7, so C = 34*log(7)/49.