New sequence : Primes for which SQRT(A000040(n)) < A001223(n)

[Dean Hickerson]

Richard Guy wrote:

> Fools rush in ...
>
> I believe that it would follow from the Riemann hypothesis.     R.

I don't think so.  Assuming RH, it's known that

  . pi(x) - li(x) = O(x^(1/2) log(x)),

which is not enough to show that there's always a prime between n^2 and
(n+1)^2.

Of course, there could be a way of estimating the number of primes
in  [n^2, (n+1)^2]  that's more accurate than just computing
pi((n+1)^2) - pi(n^2).  I haven't heard of such a thing, but I'm not an
expert in this area.

Dean Hickerson
dean at math.ucdavis.edu