In article <je2dnSC6K7IEWVbZnZ2dnUVZ_qWdnZ2d@comcast.com>, Tim Peters <firstname.lastname@example.org> wrote:
>He claims that these first few pages prove: > > |pi(n) - li(n)| < sqrt(li(n))  > for all n > 1
>Anyway, /if/  holds RH follows, because: > >a. sqrt(li(n)) < sqrt(n) for n > 1 (which he notes) > >b. so |pi(n) - li(n)| < sqrt(n) for n > 1 would follow from  > >c. RH is equivalent to the statement (which he alludes to, but doesn't > give) that there's some constant C for which: > > |pi(n) - li(n)| < C sqrt(n) log(n) > > holds for all sufficiently large n; or, IOW, > > pi(n) = li(n) + O(sqrt(n) log(n)) > >So , if true, is in fact a substantially tighter bound than required to >prove RH.
At first I thought it might be too tight to be true - but now I guess such a bound has not yet been ruled out.