- Andrey Kulsha pointed out this 2005 paper/lecture by Patrick Demichel,

who I think is an amateur, but evidently a very good one:

http://www.mybloop.com/dmlpat/maths/li_crossover_pi.pdf

http://sites.google.com/site/dmlpat2/li_crossover_pi.pdf

and it later got improved and turned into a published paper:

Yannick Saouter & Patrick Demichel:

A sharp region where pi(x)-li(x) is positive,

Mathematics of Computation 79 (2010) 2395-2405.

http://www.ams.org/journals/mcom/2010-79-272/S0025-5718-10-02351-3/home.html

Interesting. Demichel makes a computer program using high precision arithmetic, exact prime counting analytic formula, and high precision huge tables of zeta function zeros, to seek the least x such that

pi(x)>li(x). He has reason to believe he succeeded in finding that x

and it is (accurate to 10 significant figures):

x = 1.397 162 914 * 10^316

Amazing! How did he do it? Well, he has semi-empirical bounds on the error of his approximate prime-counting program. These

bounds could be wrong, but it is unlikely. He has estimates of how unlikely. (Very.) Assuming they are not wrong, he covered the whole region below this x with enough precision that he believes he's excluded it. It must have been huge amount of work. Candidate locations below this x were examined with "zoom" using higher precision, smaller spacing, and more zeta zeros, in some cases as many as 10^10 zeros, and often one needs to zoom in another level of zooming then, and so on. Anyway, he did it.

Saouter then came and was able to prove rigorous eror bounds which showed Demichel's x genuinely has pi(x)>li(x).

So that part is rigorous; the fact this x is minimal is not

since it depends on semi-empirical error estimates.

I suppose one could now ask for all 317 digits of this magic x,

but that seems out of reach with any method I can think of.

Rigorously proving this x is indeed minimal seems more likely to be possible.

Completely useless of course, but impressive testimony to the power

of analytic number theory. - _
> 3d. Re: Demichel

Shows that that dissertation did not have careful proofreading.

> Posted by: "Phil Carmody" thefatphil@... thefatphil

> Date: Sun Jan 2, 2011 4:32 am ((PST))

>

> --- On Sun, 1/2/11, Andrey Kulsha wrote:

>> There are also two new (very similar, huh)

>> papers analyzing and improving these results:

>> http://eprints.ma.man.ac.uk/1541/01/Munibah2010.pdf

>

> "In 1914, numerical evidence proved that π(x)< li(x) for all x. "

>

> Ewww...

>

> Phil

>

http://primes.utm.edu/howmany.shtml

However in 1914 Littlewood proved that pi(x)-Li(x) assumes both positive

and negative values infinitely often.

Kermit