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)
thread_title:  Early thread(s) 

message: 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)
thread_title:  Riemann's counting function 

message: Does anyone know what definition of Li is used in the Riemann-von Mangoldt formula? -- EJ 20:01, 15 January 2006 (UTC)
replies: text: Nevermind, I found it here: . It is quite a mess, actually. I'll try to fix the article later. -- EJ 18:02, 16 January 2006 (UTC)
thread_title:  Explicit formula 

message: 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)
replies: text: "Tetralectic constant" was poor OR and deleted 6 December 2006 after Wikipedia:Articles_for_deletion/Tetralectic_constant. PrimeHunter 11:42, 13 January 2007 (UTC)
thread_title:  The tetralectic who? 

message: 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 \pi(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)
replies: text: 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)
thread_title:  Lehmer and prime counting function 

message: 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)
thread_title: What about the sum of the primes less than x?

message: Although the page is not that large, JSH's apparent failure to sign his missives suggest that we move them to a subpage so we can sign them for him.  Sections 0 (most of, anyway), 2, and 4 seem to be JSH's.  I'm not saying that JSH's formulas are wrong, but they're self-published repeating multiple times, and, being 2-d recursive, not particularly notable, even if apparently fairly simple.  Any comments? — Arthur Rubin |  (talk) 16:32, 4 April 2007 (UTC)
replies: text: \pi (x) = \sum_{n=1}^{\infty} \frac{\mu (x)}{n}J(x^{1/n}) 
In fact this appears in [] but with different name (they call it f(x) instead of J(x) ) --Karl-H 18:34, 4 April 2007 (UTC)

text: The formula already appeared a few lines further down.  — Arthur Rubin |  (talk) 22:51, 4 April 2007 (UTC)

text: Nope, I was wrong.  I've re-added, and removed second reference to Möbius inversion.  Sorry about that. — Arthur Rubin |  (talk) 22:57, 4 April 2007 (UTC)
thread_title:  Archiving JSH 

message: The significance of the differences between π(x) and x / ln x, and π(x) and Li(x), depends on the magnitude of the values, and as the numbers get large it's increasingly difficult to tell by eye how significant these differences actually are. Suggest replacing with a table that in some suitable way shows the relative error. This would be more useful for getting a quick idea of how accurate the approximations are. Matt 19:55, 2 January 2008 (UTC)
thread_title:  Table of π(x), x / ln x, and Li(x) 

message: About 3/4 of the way down the page is a formula for pi(x) that has an arctan in it.  Is this formula true?  I cannot find it in the literature.  Does anyone have a reference for it?  —Preceding unsigned comment added by MathPerson (talk • contribs) 18:39, 21 April 2010 (UTC)
thread_title:  Prime Counting Function With arctan 

message: the mobius numbers satisfy sum mu(n)/n=0 and this explains why the literature defines the R-function with either li
or Li  — Preceding unsigned comment added by 2A00:1620:C0:64:21C:61FF:FE03:A4C (talk) 07:49, 16 December 2012 (UTC)
thread_title:  Might be useful? 

message: I don't think the decimal logarithm is being used at all in this article, so somebody with time should--if s/he knows the wikipedia standard--make all uses here one or the other.204.108.237.194 (talk) 20:09, 22 December 2012 (UTC)
thread_title:  'log' and 'ln' 

message: from Formula for primes Talk, someone posted this eq'n, which quickly is unruly but it seems to work.  I show in the format for Pari:
PrimePi(n) = 2+sum(x=5,n,(floor( (x!/x^2)-floor(x!/x^2)+0.3 ))).  --Billymac00 (talk) 23:46, 29 July 2014 (UTC)
thread_title:  Unusable But ... 

message: https://link.springer.com/article/10.1007%2Fs11139-011-9362-6
This is a paper with the that deals with \pi^2(x) Ramanujan formula. 
The link stated with this formula must be the wrong book. Try to check if it should be book four instead of three. I do not have them so I can not.
: The slightly later 2013 paper indicates it is in Berndt part IV.  Indeed, it is in chapter 24, pages 112-113 of that book, shown as "Entry 2 (p. 310)".  It also references the work of Wheeler ("An inequality of Ramanujan") and Keiper/Galway (no direct refernce) doing calculations related to this.  DAJ NT (talk) 21:08, 27 February 2018 (UTC)
thread_title:  Link? and new link 

message: I noticed this work  1, and while I cannot speak to the paper's quality, the formula offered seems to work well through the 10^27 entry of the article.  I just make note of it...--Billymac00 (talk) 00:23, 31 July 2018 (UTC)
replies: text: Why not just use the logarithmic integral? I mean, sure, using another term in the asymptotic expansion will help, although it's a little silly to claim that it's counting the twin primes (especially since the next term has coefficient 1, not 2). But sure, whatever approximation works for you! - CRGreathouse (t | c) 17:14, 19 November 2021 (UTC)
thread_title:  Recent work of Kristyan 

message: I add a Link to this Talk, because if you searce in German for (x) there is only a Link to the German page of Prime number theorem Steedmarks (talk) 09:33, 16 October 2018 (UTC)
replies: text: We don't link to talk pages in See also sections so I reverted it. I'm not sure what you mean but this is the English Wikipedia and our links are not determined by Wikipedias in other languages. If you mean there is no link to German under "Languages" then it is because the German Wikipedia does not have a separate article about the prime-counting function. Their article about the Prime number theorem is linked under "Languages" in that article. Two articles in one Wikipedia language does not link the same article in another language. PrimeHunter (talk) 10:09, 16 October 2018 (UTC)
thread_title:  Link to this Talk