OFFSET
1,2
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.
LINKS
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.
FORMULA
The maximum value for sigma(n)*log(n^2)/n^2 occurs at n = 7, so C = 34*log(7)/49.
EXAMPLE
1.350223368732258211705754964838124710360426138...
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]
PROG
(PARI) log(7)*34/49 \\ Charles R Greathouse IV, May 14 2012
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, May 14 2012
STATUS
approved