Talk:Divisor summatory function

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[edit] Merge

I suggest merging this article, and Dirichlet divisor problem, as these are on essentially the same problem. I did not discover that article until, of course, I'd started this. linas 05:20, 13 July 2006 (UTC)

[edit] Confusion over formulas

There is some confusion over formulas. The multitude of sources say xlogx + x(2γ − 1) in the asymptotic behaviour. However, actual numerics makes it pretty darned incredibly clear that the true asymptotic behaviour has a xlogx − 2x(1 − γ) instead. At first, I thought this was a typo in the textbook, but now I am not sure, I haven't checked. Perhaps there's an error in my source code. All this needs double checking; for the moment I left it all hanging. linas 05:20, 13 July 2006 (UTC)

Anyone with a numerics package, please check this for me. My source code passes a variety of double-checks, I am just not seeing the error, and yet the numerical behaviour seems incontrovertible, at least for N less than a million (!) ... yes, a million is a small number, (I had to compute the exponential sum on the von Mangoldt function to two billion!! terms before the divergent behaviour became clear. But really, I am quite surprised. linas 05:39, 13 July 2006 (UTC)
I have had a quick look at the standard proof of the result (xlogx + x(2γ − 1) + error) and cannot see anything wrong with it. Also checked the statement in a variety of sources, so I believe it is right despite the numerics. Since we are dealing with an asymptotic formula, I would not be too surprised to see numerical diffences at small numbers - how high can your calculations realistically go? As for the merger, I would have no problem with the proposed merger - if it is done carefully, I am sure it would improve things. Madmath789 06:34, 13 July 2006 (UTC)
Just done some calculations (10^2, 10^3, ... 10^13) and found that the xlogx + x(2γ − 1) + error version seems accurate, and (with these small values of x) the error term seems to be pretty close to a small mutlipe of x1 / 4 - even though it has been proved that the true error must be greater than O(x1 / 4). Madmath789 11:53, 13 July 2006 (UTC)
Never mind. I'd recently done a "performance improvement" and failed to run my validity tests on it. There's a blatent off-by-one error. Oh well. Numerically, 2.5*x^7/22 is a tight fit; I'll try to prepare graphs up to 10 million later. Thanks, though, for the response. linas 14:10, 13 July 2006 (UTC)

[edit] Name

Is this really the name for this concept? It seems ungrammatical, somehow. — Arthur Rubin | (talk) 17:22, 13 July 2006 (UTC)

I agree it sounds awful, but I think it really is the right name, unfortuanately. The alternative: "Summatory function of the divisor function" sounds even worse ... I prefer the title "Dirichlet divisor problem", but that would not suit the whole content of this article. Madmath789 17:29, 13 July 2006 (UTC)
There are other divisor problems besides the Dirichlet divisor problem; the Dirichlet problem is historically the first. The term "summatory function" is common in number theory, see for example Chebyshev function, with is the summatory von Mangoldt function. linas 05:21, 14 July 2006 (UTC)

[edit] Hyperbolic groups

Its not entirely clear to me whether or not the "hyperbolic simplex" bounding the points can be used to generate one or several different hyperbolic groups (in the sense of Gromov). Or is there no consitent defintion for the generators? Given such a "simplex", how many different groups may be generated? Does the divisor summatory function define a "volume" of a hyperbolic group, and does the concept generalize? p.s. beware, this is a late-night question, probably has a non-crazy answer that will become appearant in the morning. linas 05:21, 14 July 2006 (UTC) Never mind, simple examples don't work. linas 05:34, 14 July 2006 (UTC)

[edit] The merger - some details

I think you are doing a great job Linas, but I have a couple of small queries:

  • I believe that Hardy and Landau proved that \inf \theta > 1/4 not \inf \theta \ge 1/4 (needs checking).
  • On the line about Kolesnik, I think you need to interchange the 'less than' and 'less than or equal to' signs - don't you?

Madmath789 14:39, 14 July 2006 (UTC)

How about now? Actually, I should refine this so that some definitions use the epsilon, and others do not. I admit I was sloppy here. Later ... linas 15:16, 14 July 2006 (UTC)
Yep - improving all the time :-) and after some searching, I am now convinced that it really IS \ge in the Hardy Landau bit. Incidentally, I found a pdf of an interesting paper by Lioen and Lune on computational results on Merten's Conjecture and Dirichlet divisor problem - you might be interested, but I have forgotten where it came from. Madmath789 15:31, 14 July 2006 (UTC)