A tricky way to estimate pi(x)
 

It seems I've found some kind of analytic way to compute the prime counting function pi(x):

[url]http://xyyxf.at.tut.by/pi/pi.html[/url]

The feature is that there's no need to have a table of zeta zeros since the method use them, say, very marginally. Would you please take a look and say if this method is completely useless or maybe there are some future trends...?

WBR,

Andrey

I've slightly updated the page (more expressions for Nk(noise), some strange constants; it seems that (7) should be modified a bit).

On Monday I'll try to estimate the precision of the values of pi(x) obtained by my method for some x's and T's...

All the things are clear now :)

A new idea suddenly came to my mind. I mean the statement (13) here:

[url]http://xyyxf.at.tut.by/pi/pi.html[/url]
(please refresh this page in your browser because there are some additions).

Numerical computations confirm the conjectured equality. Just look how pretty is (14)!

Maybe you have some ideas how to prove (13)?

Thanks a lot,

Andrey

p (n).
 

[QUOTE=XYYXF;85382]A new idea suddenly came to my mind. I mean the statement (13) here:

[url]http://xyyxf.at.tut.by/pi/pi.html[/url]
(please refresh this page in your browser because there are some additions).

Numerical computations confirm the conjectured equality. Just look how pretty is (14)!

Maybe you have some ideas how to prove (13)?

Thanks a lot,

Andrey[/QUOTE]
:smile:
I have gone thru your URL's and honestly feel your are basing your results on too many proofs, provided that the Riemann conjecture is correct.
There are various methods now available to get p(n) which can be programmed on computers and unless yours is a short cut(which I doubt)
it will be just a theoretical curiosity.
Any way if it will be of some help to you try to procure a copy of
'Riemann's Zeta function' by H.M. Edwards-Dover ISBN 0-486-41740-9. and a companion volume of G.H. Hardy and Wright on the theory of numbers a classic in Number theory. These may be difficult in your part of the world but you may get one from the libraries to check out..
Dasvidanya! and all the best
Mally :coffee:

[QUOTE=mfgoode;85521]your are basing your results on too many proofs[/quote]The only proof I need is the proof of (13) :smile:

The Riemann Hypothesis is not really needed, we can just add -R(x^r) for Re(r)<>1/2 to our sum.

[QUOTE=mfgoode;85521]There are various methods now available to get p(n) which can be programmed on computers and unless yours is a short cut(which I doubt) it will be just a theoretical curiosity.[/quote]But all these methods were found theoretically. Well, an interesting curiosity may also have some future trends :smile:

[QUOTE=mfgoode;85521]'Riemann's Zeta function' by H.M. Edwards-Dover ISBN 0-486-41740-9. and a companion volume of G.H. Hardy and Wright on the theory of numbers a classic in Number theory. These may be difficult in your part of the world but you may get one from the libraries to check out..[/quote]Of course I have free access to these books, as well as to Titchmarsh's one and some others. BTW, (15)-(19) heuristically follows from theorem 14.21 in Titchmarsh1951.

[QUOTE=mfgoode;85521]Dasvidanya! and all the best
Mally :coffee:[/QUOTE]:smile: :smile: :smile:

[QUOTE=XYYXF;85608]The only proof I need is the proof of (13) :smile:

The Riemann Hypothesis is not really needed, we can just add -R(x^r) for Re(r)<>1/2 to our sum.

But all these methods were found theoretically. Well, an interesting curiosity may also have some future trends :smile:
:[/QUOTE]

:rolleyes:
Well XYYXF I hope your proof of (13) will be more than a curiosity, one can never tell!
After that you should send in a paper to the AMS for checking it out and verification, plus they will officially acknowledge it as your paper.
It looks like we have another Grigori Perelman in the making!
Cheers! Russky devushka ochin kharasho! ochin,ochin! Scold! :wink:
Mally :coffee:

[QUOTE=mfgoode;85666]Well XYYXF I hope your proof of (13) will be more than a curiosity, one can never tell![/quote]I'm trying hard :smile:

[QUOTE=mfgoode;85666]After that you should send in a paper to the AMS for checking it out and verification, plus they will officially acknowledge it as your paper.[/quote]Whoops :rolleyes:

Don't think the result is too important... but If I succeed to estimate (heuristically) pi(10^23) or something around, and the value is confirmed by traditional method, this will maybe raise the interest from people...

[QUOTE=mfgoode;85666]It looks like we have another Grigori Perelman in the making![/quote]Perelman is much much much much more earnest mathematician than a number theory enthusiast Andrey Kulsha :blush:

[QUOTE=mfgoode;85666]Cheers! Russky devushka ochin kharasho![/QUOTE]Devushka means a young female, but I'm male instead :smile:

I'm not entirely sure I understand your method, but let's see how it works. Can you calculate your estimate for small values? pi(10^10), pi(10^11), pi(10^12), and so on?

That it what I'm doing now, but first of all we need some pre-calculated values of Nk(noise)(T). I've chosen

T = 1370919909931995308285.4

The ball is rolling... :)

Well. I realized that we need to specify the values of zeta zeros number 10^22 to 10^22+10^3 to, say, 200 decimal digits.

[url]http://www.dtc.umn.edu/~odlyzko/zeta_tables/[/url]
[url]http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros5[/url]

These values are accurate to within 10^6, I'll try to get more digits with Mathematica 5.0...