Talk:Logarithmic integral function

Starts at?

Do you want Li(x) to be the integral that starts at 0, or the integral that starts at 2? You use two different definitions in this article. Furthermore, for which values of x is Li(x) defined? AxelBoldt, Tuesday, April 2, 2002

See renewed article for improvements. I know it is not perfect. I still have to do more on a definition range of Li(x). Please be patient :-) Perhaps nice graph would solve the problem? Your futher remarks are very wellcome. And we should start to correct also Eric Weisstein's math pages because he almost does not use any definition ranges or does he? --XJam [2002.04.02] 2 Tuesday (0).

No, he often glosses over these things. Yes, a graph of li(x) would be nice. I added the ranges and corrected some formulas. I have a question about:

π(n) = ∑_m=1^∞ μ(m)/m ∫_μⁿ 1/ ln t dt,

where μ(m) is Möbius function.

Where did you find this formula? AxelBoldt

You can find this form of prime number theorem at http://mathworld.wolfram.com/SoldnersConstant.html.

I don't trust that formula; it looks wrong to me. Weisstein has lots of mistakes. Can you find it somewhere else?

This is a hard field and I doubt if we can check a formula at a glance. I'll go finding it.

This is Ramanujan's result and Ramanujan was known that he did not understand math in full, he, he. The same thing happend when Hawking corrected Zel'dovich and Starobinsky's work on Hawking radiation of black holes or something like that not to speak about Feynman. Who blames Gauss that he was too young to give a second estimation for π(ξ)? I would like also to clarify some of the following things if it is posible.

ÄCHTUNG (= outlawry (Small word game on famous German attention))

1// Nice you've put some definition ranges in equations. As I am partly physicist it shows that I am not as rigorous as mathematician - but I think this is not some special fault. At first you argued about two different definitions of li(x) and Li(ξ) and afterward you had thrown out some further explanations, mainly concerning famous Li(ξ).

I saw somewhere that li(x) and Li(x) are used interchangeably, and so I didn't want to confuse the issue. The "logarithmic integral" as such is always the integral that starts at 0, with Cauchy principal value. We can even use this version in the prime number theorem; it is slighly more accurate than using the integral that starts at 2.

2// If a function is non elemental - isn't this important? Should we searh for this property in appropriate article function. I doubt there's an explanation for this. (I haven't checked).

Yup, I put that back in.

3// I've noticed you don't like mathematical terms to be named in two or more ways (i.e. Euler constant and so on). What approach should we use here? You left for example opened naming Ramanujan-Soldner constant but mostly is used Soldner's one.

If there are several names for a concept, you describe those several names on the concept's page, but not on every page that links to the concept.

Yes this was my intention because one still have to do a link to that particular constant if that is desired.

4// It is nice to know that some terms in li(e^u) and li(x^m) must have absolute values (i.e. ln| ln x|). Where can we find this strictly terms of this kind of series?

It was mistake in Weisstein. You need the absolute value.

I've got them from Bronstein and Semendyayev not from Eric's sources... But I must add that many pure mathematicians give links and references to Eric's pages and they are not wrong all I guess. I like a form of that pages (do you know anyone better). Wikipedia is still in nappies for math.

5//> For x > 1 in a point t=1 this integral diverges. < What is wrong with this one? Yes, this is just main definition range. It is not a definition range in full, because an integral is defined also for 0 < x < 1.

Nothing wrong with this. It's still there, just a bit reformulated.

6// OK. We put out the entire connection with Li(ξ) and π(ξ). Connection lost. We must see prime number theorem and back forth. But why we have to throw away this Li(x) = li(x) - li(2)? This is an important property of Li(ξ), isnt it? This is also important > logarithmic integral is defined with no Cauchy's principal value so that Li(2) = 0 <

As I said above, we can use li(x) in the prime number theorem (which is what Gauss did in fact).

Yes, but we are not all Gausses. This is fine particular case. I have seen mostly Li(ξ) in number theory. Both integrals do not map π(ξ) in a whole, so there's only a small difference - but it is good to know it.

7// I've written > Thus we can rewrite Cpv ∫₀^x 1/ln t dt by ∫_μ^x 1/ln t dt in x>1. < I think this is very wide-ranging Ramanujan's adoption and I don't know why this can't be in article... You just put another definition for Cauchy's principal value.

My version of the article explains what the Cauchy principal value is, while your version simply says "take the Cauchy principal value" without explaining it.

But - as you said - this must be explained in other article (concept's page)! Yes, this was mentioned again to put related stuff out there but for first view is good to have with. In that way I am freely to go and write Ramanujan-Soldner-Stohacky's constant everywhere I want and back on to ∞. I have a question about this. How should be articles in Wikipedia be atomized or be structured? Should they be like Cantor's dust or should they look like a monster group? Think on physics. If atoms and 'superstrings' or even numbers were slightly different, we wouldn't be here to argue about their structures and such. What to explain and what to mentioned? I prefer more things to mention than nothing or few.

8// I was explaining nothing else but terms li(x), Li(ξ), ∫_μ^x with Möbius function μ(n) or with prime number theorem. I would like to put these things back in the article but the thin red line which separates an article from encyclopedia's term is far from my sight...

As I said above, I don't trust this formula. I'm sure there is a formula like that by Ramanujan, but Weisstein's version is wrong.

9// Nobody didn't tell me I can't use binary 'semi' operator ≡ in Wikipedia... Nobody told me I can't bold terms as li(x), Li(x) in definitions. Someone told me (I think it was Zundark) we can't use colors in Wikipedia.

Why do you want ≡ instead of the simple "="?

Because it is quite informational. It shows an equation to be a definition. Think on pascal-like unitary operator for setting values to variables (:=). If you think that we don't need ≡, we simply don't need it.

10// I have a lot of will for mathematics and related things but now I am really afraid to say anything that I know or that I don't. I still have some things to say but I'm too tired to do so. I would like to know what would be if Ramanujan would live longer or if he and Gauss would have personal computers. (Perhaps I'll deside to stick with plain TeX and native stupidity) :-) --

11//Axel, is anything of above posible. Do I need any confirmation from Wikipedian collegiate body? I would like to hear someone else's opinion too. What is really worth if someone (you) corrects the understanding of something for 20% but he reduces it by 21%. This is like going two steps forward and one backward. Chaos and blirt. --XJam [2002.04.03] 3 Wednesday (0).

Why do you always get so irritated about every little thing of you I edit? This editing of each other's material is the name of the game after all. AxelBoldt

I guess I have to get used on it. Another question comes to me about this. :-) I know Germans are very precise people. It is not commonly known that Slovenes are too. :-) (Pst, perhaps even more than Germans, ha, ha, but they are outnumbered). I wonder about another property of human activities. Which nation is the most methodical or systematic one? It would be nice to know. If there is one in the first place. I am not irritated (perhaps it looks like). I just have my own way of putting things outta me. And I know I am not alone. We are not equal at all that is for shure. And to paraphrase Joseph Hill in the end: there are three sides of mathematical truth... Best regards. --XJam [2002.04.04] 4 Thur's day (0).

Why does the formula for the growth behavior for x->\infty use \Theta rather than O? Wilke 19:14, 11 May 2004 (UTC)

I moved this here and made logarithmic integral a disambiguation page since there is an entirely different meaning to the word. Gene Ward Smith 02:44, 16 Feb 2005 (UTC)

Math noob question: can the logarithmic integral be evaluated in closed form? I would like to see this question answered in the article. Thanks! Greg (theytsejam) 20:25, 17 June 2007 (UTC)

Should there be a bit more on the connection between this function and the PNT? For example, the error difference, and Littlewood violations? My 2 Cents' Worth (talk) 13:24, 21 February 2011 (UTC)

ali(x)?

I was looking for the the inverse function of the logarithmic integral function, ali(x). Is there a reason that it is not mention here? John W. Nicholson (talk) 01:39, 12 April 2014 (UTC)

Li (x)-pi (x)=O(sqrt {x}\log x)

This bound is referenced from Abramowitz and Stegun. The new digital version DLMF dated 12/15/2019 shows the best bound as:

|π(x)−li(x)|= O(x *exp(−d(lnx)^3/5 * (lnlnx)^(−1/5))).

I have no familiarity otherwise so I cannot comment.--Billymac00 (talk) 03:44, 28 December 2019 (UTC)

Does this belong here?

The logarithmic integral is defined as either ("American" definition starting at 0, for which there is a singularity at $x$ = 1)

$\operatorname {li} (x):=\lim _{\eta \to 0^{+}}\left(\int _{0}^{1-\eta }+\int _{1+\eta }^{x}\right){\frac {du}{\log u}},$

or ("European" definition starting at 2)

$\operatorname {li} (x):=\int _{2}^{x}{\frac {du}{\log u}},$ 49.185.41.241 (talk) 12:45, 29 June 2022 (UTC)