One of the problems, I think, that you might bump into is that many of the interesting combinatorial prime counting algorithms out there (like the method djbroadhurst linked to) don't seem to have any obvious connection to the logarithmic integral, so it's hard to know where you would add the error terms back in.

In addition to dj's links, if you're really interested in playing around with this approach, one set of ideas you might take a look at is this:

http://www.icecreambreakfast.com/primecount/logintegral.html
(warning: I just wrote this up last week, so consider this un-peer-reviewed, suspect, and possibly self-aggrandizing) It actually has some reasonable raw material for an attempt at what you're talking about, I think.

Nathan McKenzie

--- In primenumbers@yahoogroups.com, "techno.buddhist" <yahoo@...> wrote:

>

> Hi all,

>

> Newbie with a newbie question here. Other than Riemann Hypothesis is no other way to calculate 'exact' P(x)?

>

> If there was such a method then somebody could do the opposite of Riemann and work from the solution by adding waves to get to x/Li(x)?

>

> In other words prove RH by downgrading the exact result to meet the PNT estimate of x/Li(x)?

>

> Thanks

>

> TechnoBuddhist

>