Talk:Noncototient

improving math definitions in wikipedia

many authors of wiki are poor in explaining, where their definition is so aloof that only another math graduate understands them. The case of noncototient is an example where a high school or undergraduate would not understand what cototient means because the author is so opaque in explaining. If the author were to show in detail how 8 and 9 were cototient that a High School student could understand then the author has done a good job instead of this lousy job. AP

Density

Which definition of density is meant by the concluding phrase, "positive lower density"? The natural density? The Schnirelmann density? Some other density? Should there be a link on that phrase?

That phrase was added by 140.180.164.252 in the second revision of the article long ago, so there is no direct way to ask him/her which one was meant.

Therefore, it's up to us to review that statement. That means reading the papers of "Browkin and Schinzel (1995)", "Flammenkamp and Luca" and seeing if we can find anything about that. If you like, you can remove that statement and put something like "Removed density statement pending review (see Talk)". PrimeFan 22:34, 10 March 2006 (UTC)[reply]

I'm not prepared (academically) to read such papers. However, I did find this definition of asymptotic density, upper asymptotic density and lower asymptotic density; it gives "lower density" as a symnonym for "lower asymptotic density". The definitions appear to be similar to natural density (which asymptotic density redirects to). Is my feeble mind close to correct?

Sequence density as defined by Mathworld [1] appears to be a very similar concept. But until someone can say for sure which density is referred to and explain it to the others, I've decided to take Primefan's advise and remove the statement. Anton Mravcek 22:04, 11 March 2006 (UTC)[reply]

A paper of Banks and Luca at Arxiv math.NT/0409231 shows that the number of noncototients less than X is bounded below by (1/2 + o(1))X/log X [which might well be added to the article --- the Flammenkamp--Luca result gives only a constant multiple of log X as lower bound]. The natural question is thus to ask whether the set of noncototients has positive lower density in the asymptotic sense: that is, whether there is a constant c>0 such that the number of noncototients less than X is infinitely often greater than cX.

Thanks for the link to the Banks & Luca paper. All of us here ought to read it closely and reflect on it. PrimeFan 23:04, 25 July 2006 (UTC)[reply]

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