[seqfan] Re: Comment in A003418

[Charles Greathouse]

If |psi(x) - x|/(sqrt(x) log^2 x) is unbounded (consequent on not-RH)
than there is some x for which |psi(x) - x|/(sqrt(x) log^2 x) > 1000
and hence |psi(x) - x| > 1000sqrt(x) log^2 x. That gives one
direction. In the other direction, if RH is true then |psi(x) - x| <
(1/(8 pi)) sqrt(x) log^2 x < 1000sqrt(x) log^2 x.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Sep 13, 2012 at 9:52 AM, Georgi Guninski <guninski at guninski.com> wrote:
> On Thu, Sep 13, 2012 at 09:30:13AM -0400, Charles Greathouse wrote:
>> 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.
>>
>
> I don't get it.
>
> If you replace "sqrt(x) log^2 x" by "1000 sqrt(x) log^2 x" in the
> comment the equivalence will fail no matter if your formula is bounded or not.
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/