## Demichel

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• Andrey Kulsha pointed out this 2005 paper/lecture by Patrick Demichel, who I think is an amateur, but evidently a very good one:
Message 1 of 8 , Jan 1, 2011
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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

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.
• ... And one with access to some fairly impressive computational clusters. Phil
Message 2 of 8 , Jan 2, 2011
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--- On Sun, 1/2/11, WarrenS <warren.wds@...> wrote:
> Andrey Kulsha pointed out this 2005
> paper/lecture by Patrick Demichel,
> who I think is an amateur, but evidently a very good one:

Phil
• ... There are also two new (very similar, huh) papers analyzing and improving these results: http://eprints.ma.man.ac.uk/1541/01/Munibah2010.pdf
Message 3 of 8 , Jan 2, 2011
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> 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

There are also two new (very similar, huh) papers analyzing and improving these results:
http://eprints.ma.man.ac.uk/1541/01/Munibah2010.pdf
http://eprints.ma.man.ac.uk/1547/01/SZproject2010.pdf

Best regards,

Andrey

[Non-text portions of this message have been removed]
• ... In 1914, numerical evidence proved that π(x)
Message 4 of 8 , Jan 2, 2011
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--- 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
• ... It gets worse. Looking at the start of Chapter 3, Numerical Results (i), we have the assertion: Now we know that (gonna simplify glyphs, sorry) e^(iwy)
Message 5 of 8 , Jan 2, 2011
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--- On Sun, 1/2/11, Phil Carmody wrote:
> --- 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...

It gets worse. Looking at the start of Chapter 3, Numerical Results (i), we have the assertion:

Now we know that (gonna simplify glyphs, sorry)

e^(iwy) e^(iwy) e^(-iwy)
------- = ------- + --------
p B + iy B - iy

where none of the terms are defined. It seems chapter 2 most recently defined p = B + iy.

So his assertion seems to be that:

e^(iwy) e^(iwy) e^(-iwy)
------- = ------- + --------
B + iy B + iy B - iy

Or:

e^(-iwy)
0 = --------
B - iy

Or are my eyes playing tricks with me?

Phil
• ... I guess they mean the conjugated pair of zeta zeros. Best regards, Andrey
Message 6 of 8 , Jan 2, 2011
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> It gets worse. Looking at the start of Chapter 3, Numerical Results (i),
> we have the assertion:
>
> Now we know that (gonna simplify glyphs, sorry)
>
> e^(iwy) e^(iwy) e^(-iwy)
> ------- = ------- + --------
> p B + iy B - iy

I guess they mean the conjugated pair of zeta zeros.

Best regards,

Andrey
• ... Maths by guesswork? What happened to rigour? Then again, seeing how long those two take to get from 24/8 to 3, I suspect that the papers will be closely
Message 7 of 8 , Jan 2, 2011
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--- On Sun, 1/2/11, Andrey Kulsha wrote:
> > e^(iwy)    e^(iwy)   e^(-iwy)
> > ------- =  ------- + --------
> >    p        B + iy    B - iy
>
>     I guess they mean the conjugated pair of zeta zeros.

Maths by guesswork? What happened to rigour? Then again, seeing how long those two take to get from 24/8 to 3, I suspect that the papers will be closely associated with rigor mortis.

Phil
• _ ... Shows that that dissertation did not have careful proofreading. http://primes.utm.edu/howmany.shtml However in 1914 Littlewood proved that pi(x)-Li(x)
Message 8 of 8 , Jan 2, 2011
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_
> 3d. Re: Demichel
> 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
>

Shows that that dissertation did not have careful proofreading.

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
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