[seqfan] Re: Comment in A003418

[Charles Greathouse]

I believe the original comment is correct: if RH fails then |psi(x) -
x|/(sqrt(x) log^2 x) should be unbounded. (It's certainly unbounded if
there are infinitely many zeros off the critical line; can anyone
address the case with only finitely many?)  But it would be good to
include the Schoenfeld improvement, of course.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Sep 13, 2012 at 3:23 AM, Georgi Guninski <guninski at guninski.com> wrote:
> https://oeis.org/A003418
> a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, ..., n}
>
>
> a[x]=exp(psi(x)) where psi(x)=log(lcm(1,2,...,floor(x))) is the Chebyshev function of the second kind.
>
> Not sure if this comment is correct:
> An assertion equivalent to the Riemann hypothesis is: | log(a(n)) - n | < sqrt(n) * log(n)^2 (for n>=3).
>
> RH implies:
>
> |psi(x)-x| < sqrt(x) log^2 x / (8 pi) if x>73.2
>
> Is the constant $8 pi$ missing from the comment or the comment is correct?
>
>
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