Talk:Prime counting function

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This is a very strange stub. The format does not fit very well with other math articles, the style is rather poor, and there is an accuracy issue as well (this is what has prevented me from expanding the stub). If I am reading everything correctly, the "recent" discovery is a modified/obfuscated Legendre-Meissel recursion (we're just counting composite numbers by inclusion-exclusion). Suggestions?

Alodyne 01:06, 6 Dec 2004 (UTC)

Edited to handle points of dispute.

This unusual and not terribly effective formula is the work of sci.math denizen James S. Harris, who has produced a large body of work of dubious quality. This morning, he wrote (on sci.math):

As an experiment I put my prime counting function on the Wikipedia writing the entire article. You could see how a concise prime counting function fits nicely in an encyclopedia artice, and I saw it as an opportunity to see if the math world could behave at all like expected.

The usual dispute with James Harris is over the quality and relevance of his work. The quality or relevance is, however, quite irrelevant to its appropriateness here, since its the manner of its appearance is directly contrary to Wikipedia policies (specifically, the policy against self-promotion).

That said, having a page on prime-counting functions (with, for instance, Legendre's method) is a good idea. An internal link to the Prime Number Theorem would be worth having, but redirecting seems misleading, since the Prime Number Theorem provides only an approximate count. --Jake 18:18, Feb 1, 2005 (UTC)

Hey, I have an unusual situation in that mathematicians are refusing to acknowledge a correct and interesting mathematical result, which also happens to fit easily into a relatively small encyclopedia article. Notice the disparagement in Wildstrom's comments, when the formula itself is just math.

I emphasize that Wildstrom felt it necessary to disparage. I'm not dealing with people who are behaving objectively from the math world, but they are very emotional on this subject and refusing to just acknowledge a correct mathematical result. It's personal for me as the discoverer, but it remains to be seen why it's so personal to them.

I have what is an actual prime counting function. It is easily derived, but I derived it, and apparently mathematicians have decided for that reason they will never acknowledge it.

I plead a special case here as look at the current article and consider how abstruse it is, versus my article.

Also note that no one not an expert in the field could actually count primes with the current article, but my formula allows a non-expert in the field to actually do so, plus I give the additional information about the prime number theorem and current research.

I say that mathematicians are just being weird here, and obviously, if mathematicians refuse to acknowledge a correct and easily proven result, then I run into a problem putting up a citation!

JSH 18:56, 5 Feb 2005 (UTC)

I've added a normal sort of prime-counting function page, which I may expand on. Gene Ward Smith 02:26, 3 Feb 2005 (UTC)

Consider, the page was reverted back to the redirect and then there was no prime counting function article for a while, until after I mentioned my experiment on the sci.math newsgroup THEN someone came to try and write an article versus just leaving the re-direct to the prime number theorem.

Now compare their abstruse, hard to understand prime counting function article, which does not contain a formula that a non-expert can use, with my own, and the real issue here I think is a bizarre case of mathematicians fighting to not acknowledge a correct result.

The rules limiting to cited research don't handle a situation where the experts are not behaving as expected with an easily derived math formula, where correctness is not in doubt.

JSH 18:56, 5 Feb 2005 (UTC)

I think the article needs to have the methods of getting an exact count on prime numbers (other than a brute-force sieve), or either a link to a page that has that information. Bubba73 22:05, 21 Jun 2005 (UTC)

Most of the methods for counting prime numbers, especially the ones actually used, are too involved for an encyclopedia article. My prime counting function is distinctive in that it's short enough for an article, while it is slow, though a fast algorithm derived from it is even smaller, so it is small enough with a fast algorithm to fit in an article.

But my research isn't published in a math journal, so I'm not supposed to put it up. And don't think I could put it up anyway without people coming out of the woodworks to tear it back down as math people are VERY active in fighting my research, mostly relying on personal attacks, like you can see earlier on this talk page.

The real math field is nothing like what many of you probably think it's like, as people are very committed to promoting their careers and academics see amateurs like myself, especially in very high profile areas like prime numbers as not helpful to their own careers.

For instance, I had a paper on a different subject though still number theory published in a peer reviewed electronic math journal and people from sci.math got together and mounted an email campaign against it and the journal folded, pulling my paper, which they could try to do being an electronic journal, though it is unheard of.

So I have the message loud and clear from the mainstream math world which is they will not accept my work despite it being correct, and the rest of the world does not matter.

It's my math, as far as they're concerned, and what's my math is to be rejected without regard to its mathematical importance.

Of course, it's annoying to me, but the world pays the price, though someday I guess, as has happened with past research fought by some group or other it will be "discovered" and I'll just be another story to add to a long list of people with major results who had to fight against the established "experts" who saw their efforts as a threat.

Nothing changes in this area. People don't learn from history.

JSH 21:21, 27 August 2005 (UTC)


What amazes me is how easily math people get away with dumb crap, like writing an article on counting prime numbers that doesn't actually show you known methods to count prime numbers.

The authors of this article just glancingly mention "arithmetic" methods only go off on a tangent talking about work related to the Riemann Hypothesis.

For those who want to see actual "arithmetic" prime counting formulas see the much better article at MathWorld:

http://mathworld.wolfram.com/PrimeCountingFunction.html

I've sat and waited and waited, hoping that someone from the math community would step up and do the service of writing a real article on prime counting for the Wikipedia, but the math community doesn't care.

Compare the MathWorld article to Wikipedia and see what I mean.


JSH 04:10, 13 September 2005 (UTC)


Oh, and hey, I think it's worth it to compare both articles to my own reverted article:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&oldid=9142249

My article also talked about figures from the history of prime counting, like Gauss and Chebyshev while the current Wikipedia article acts as if prime counting started in the late 1800's with Riemann!

It's bizarre how bad that article is--like, not even mentioning Gauss??!!!--and how the community here is tolerating that, especially given that it was a reaction article from Usenet, when I talked about having written the first prime counting function article for Wikipedia, as before there was a re-direct.

My guess is that some college students fixated on the Riemann Hypothesis as it's the big thing now wrote the article, and being kids they don't appreciate, or maybe even don't know the rich history in this area, which the interested reader can get more of at the MathWorld article, or hey, at mine.

And them not mentioning Euler or Chebyshev is just amazing to me, as Euler is the one who presented the zeta function, and Chebyshev is the one who did the early limit work that others built upon, so these students--assuming they were--might not have been doing in-depth studies OR their teachers aren't doing their jobs.


JSH 15:43, 25 September 2005 (UTC)

And finally the article has been updated!!! So a lot of my past criticisms no longer apply.

Intriguingly though, some of the updates include an interesting sieve function which is, it turns out, very closely related to my own prime counting function, and looks a lot like one key piece of it.

In any event, I'm just happy a fuller article was finally written.


JSH 21:31, 13 November 2005 (UTC)


Contents

[edit] Riemanns counting function

In the section on other counting functions, J(x) is given as

J(x) = \sum_{n=1}^\infty \pi(x^\frac{1}{n})

shouldn't that be

J(x) = \sum_{n=1}^\infty \frac{1}{n}\pi(x^\frac{1}{n})

? --Monguin61 01:52, 21 December 2005 (UTC)

[edit] Explicit formula

Does anyone know what definition of Li is used in the Riemann-von Mangoldt formula? -- EJ 20:01, 15 January 2006 (UTC)

Nevermind, I found it here: [1]. It is quite a mess, actually. I'll try to fix the article later. -- EJ 18:02, 16 January 2006 (UTC)


To me the experiment all along has been to see if information in and of itself has a chance to get picked up and acknowledged.

It does not. It has to be promoted. It has to be pushed forward against resistance.

You need to be a politician as well as a discoverer or you're just waiting and hoping until some politician picks up your ideas and promotes them, which is a lesson from history as well.

Ideas don't get picked up because of their inherent value, but their value is only seen by groups as a result of the efforts of some people to promote them.

Politics is a crucial part of the researchers' toolkit.

Moving forward as I look to more social ways to make the information enticing, and to excite the imaginations of people about that information, I want it part of the record that knowledge in and of itself, is not enough.

The reality of getting results known is a political process. You have to promote, campaign, and figure out some way to excite people about the information you have, no matter how huge it is.

I know how big my find is, and how much ground in prime research it covers, but none of that matters to the simple ability of most people to just not notice.

That's the real lesson here, and I think that ends my experiment.


JSH 18:06, 8 April 2006 (UTC)

[edit] Comparison Plot

here is a comparison plot (ratios) of n/ln(n),Riemann pcf,actual and the approx found on main page (round[exp(x-A*z+B)])

[2] Comparison plot

Now, a power fit to the first 3 significant digits provides the following approx:

            #primes ~ 504.47/(logN**1.0434)      [w/ user adding the appropriate power of 10]

Thus, for N=23, the eqn returns 19.1 with E20 implied

This approx uses data for n>=10^6 and returns a value within 1% of the actual.

Realizing that the denominator power likely is tending towards unity, and reworking gives

            #primes ~ 1022/lnN

This approx returns a value within 1% of the actual for n>=10^14.--Billymac00 20:57, 20 June 2006 (UTC)

[edit] Questioning controversy

With natural numbers x and n, where pi is the i_th prime then


P(x,n) = x - 1 -\sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))}

where if n is greater than the count of primes up to and including \sqrt{x} then n is reset to that count.

P(x,n), with n equal to the count of primes up and including \sqrt{x} is the count of prime numbers up to and including x and that is just a simple formula that can be related to what was previously known in the field. More compact than anything else similar in the prime counting area.

The fully mathematicized form of the prime counting function shown above, where you don't find find a list of primes first--because the formula finds the primes on its own--is, if y\le\sqrt{x} then

P(x,y) = \mathrm{floor}(x) - 1 -\sum_{k=2}^y {((P(x/k,k-1) - P(k-1,\sqrt{k-1}))( P(k,\sqrt{k}) - P(k-1,\sqrt{k-1})))}

else P(x,y) = P(x,\sqrt{x}).

Either way gives the same count of prime numbers, of course, but the fully mathematicized form doesn't need to be given a list of primes up to \sqrt{x}, which is how this formula is crucially different from what was previously known in the field.

So you get simplicity one way--compare with prime counting functions on the main page--and complexity another way that does something never seen before, and that should add up to mathematical interest for the math community.

Yes, I've contacted mathematicians including mathematicians leaders in the field in this area, and just any mathematician I could try to get to pay attention and even none mathematicians. I've also tried to get these results published in a mathematical journal, to no avail.

Turning to the web and Usenet I've gotten ridicule which has been sharp, personal and often cruel with more energy directed the more I question the current math community's behavior, and there seems to be no way to stop them. It's like they have absolute power and are willing to use it to block whatever suits them to block.

When before I would have assumed that math was just of so much interest to members of that community that they embraced new results, whatever the source.

And you can see similarities to what was previously known but it IS different in key ways. However, I discovered that formula a few years ago and ran into a wall from the math community simply refusing to properly acknowledge or record it. They behave like they own math and attack people they see as interlopers, refusing to acknowledge information from outsiders and when you have people breaking rules in such a wacky way, what can you do?

I tried writing a page that showed the more complex form of that equation, which is the fully mathematicized version and you can see at the top of this discussion page how well that went over, but hey, I'm desperate. Math people are just saying no, refusing to acknowledge a simple formula that counts primes that is different in special ways from what was previously known and there seems to be nothing I can do about it. JSH 02:37, 8 November 2006 (UTC)

James, as your humble archivist, you know I cling to your every word. So you make it really difficult when you start flinging your words around the internet like this. Besides, it is clear that this talk page is not an appropriate forum. It's for discussing editorial decisions regarding the article, not for drumming up support for original research.
Come on now. Let's return to our usual settings. Thanks much. Phiwum 03:26, 7 November 2006 (UTC)


But, as to your formula, my favorite part is the "floor(x)" where x is a natural number. Shouldn't that be "floor(ceiling(abs(x+1-1)))"? Or, I guess x would also work since it's a nonnegative integer and floor, ceiling, absolute value and adding and subtracting 1 each all do nothing to it.

[edit] The tetralectic who?

An unregistered editor added a link to the Tetralectic constant page. That page gives a "famous" result prove in 2006. The only references are to Yahoo search results, which I did not wade through. Google comes up empty for both "Tetralectic constant" and "Tetralectic theorem". Is this a real result? If so, maybe someone can clean up that very young article. If not, perhaps a CfD is in order. Phiwum 15:11, 30 November 2006 (UTC)

"Tetralectic constant" was poor OR and deleted 6 December 2006 after Wikipedia:Articles_for_deletion/Tetralectic_constant. PrimeHunter 11:42, 13 January 2007 (UTC)

[edit] Lehmer and prime counting function

I'm not quite sure that this is relevant - but I'll post it anyway. I seem to remember (from reading one of the Berndt/Ramanujan volumes) that Derrick Lehmer's values for π(x) were out by unity - because (for some reason best known to himself) Lehmer took unity itself to be prime. Does this have any relevance to his contributions? Hair Commodore 19:22, 12 January 2007 (UTC)

A few yrs ago I chatted with an analytic # theory professor with a PhD from Berkeley, and I asked her about Lehmer: "Is he the guy who always said 1 is prime?" She smiled and said "1 is prime." Everyone laughed and somebody said 'It's a Berkeley thing." --I sometimes wonder if the analytic # theorists have some reason for wanting 1 prime to make some formulas or theorems in their area work better. Of course, 'regular' # theory wants 1 to be neither prime nor composite to make the Fundamental theorem of arithmetic work.Rich 05:49, 15 March 2007 (UTC)

[edit] What about the sum of the primes less than x?

Does anyone know what work has been done on that? And sum of squares of primes less than x, etc? (Analogy with divisor functions) Perhaps Wikipedia could have an article on it.Rich 05:22, 15 March 2007 (UTC)

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