OFFSET
1,3
COMMENTS
According to Rosser and Schoenfeld (1961), the second Chebyshev function psi(n)=log(lcm(1,2,...,n)) ~ n. Consequently, the function log(lcm(1,2,...,n))/n tends to 1 as n tends to infinity, however it has a maximum value of 1.03882... when n=113. In precise terms this constant is log(955888052326228459513511038256280353796626534577600)/113 and it provides an upper bound for log(lcm(1,2,...,n)) <= log(955888052326228459513511038256280353796626534577600)/113*n for all n>0.
LINKS
J. Barkley Rosser, Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94
Eric Weisstein, Chebyshev Functions.
FORMULA
C = log(955888052326228459513511038256280353796626534577600)/113
EXAMPLE
1.0388205776091298930081555627382465269336112084545034825058980...
MATHEMATICA
table=Table[Log[LCM @@ Range[n]]/n, {n, 1, 1000}]; max=Max[table]; n=1; While[table[[n]]!=max, n++]; Print[N[max, 100], " at n = ", n]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, May 07 2012
STATUS
approved