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386:) 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.
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668:). 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.
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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
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In section
Definition, the letter k has multiple meanings. It's one of the variables the second sum carries over, it denotes the number of numbers involved in a product, and in the end it's once again used to carry over a sum. If no one objects, I will eventually fix this.
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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.
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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.
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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.
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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.
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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
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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 ...
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Its not entirely clear to me whether or not the "hyperbolic simplex" bounding the points can be used to generate one or several different
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in the asymptotic behaviour. However, actual numerics makes it pretty darned incredibly clear that the true asymptotic behaviour has a
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On the line about
Kolesnik, I think you need to interchange the 'less than' and 'less than or equal to' signs - don't you?
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to two billion!! terms before the divergent behaviour became clear. But really, I am quite surprised.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Yep - improving all the time :-) and after some searching, I am now convinced that it really IS
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Just done some calculations (10^2, 10^3, ... 10^13) and found that the
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There is some confusion over formulas. The multitude of sources say
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the name for this concept? It seems ungrammatical, somehow. —
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I have had a quick look at the standard proof of the result (
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279:{\displaystyle x\log x-2x(1-\gamma )}
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