[seqfan] Re: Strange comment in A003418
Peter Luschny
peter.luschny at googlemail.com
Sun Oct 23 12:20:43 CEST 2011
GG: A003418 a(0) = 1; for n >= 1, a(n) = least common multiple (or
lcm) of {1, 2, ..., n}
GG: An assertion equivalent to the Riemann hypothesis is:
GG: | log(a(n)) - n | < sqrt(n) * log(n)^2. - Lekraj Beedassy [..], Aug 27 2006
GG: According to my computations the inequality fails for n in {1,2}.
log(lcm({1,..,n})) is studied under the name of psi(n) ever since Chebyshev
did so in 1852 [1]. Under the Riemann Hypothesis for n >= 74
| psi(n) - n | < sqrt(n) log(n)^2 / (8 Pi).
This was proved in 1901 by Helge von Koch [2] using an unspecified constant
in place of 1/(8 Pi). This constant was proved by Lowell Schoenfeld as
far as I know.
[1] P. L. Chebyshev, Mémoir sur les nombres premiers., J. math. pures
appl. 17, 1852.
[2] Helge von Koch, Sur la distribution des nombres premiers, Acta
Math., v. 24, 1901.
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