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June 8[edit]

Correspondence Between Behaviour of Series and Density of Sequences[edit]

This is not my area of expertise, so I apologize if this is naive. Anyways, obviously that there are infinitely many primes follows immediately from the divergence of the sum of their reciprocals. In the same vein, that the sum of reciprocals of twin primes converges implies that if they are infinite, they are less dense. Obviously, there is a link between the density of a sequence of naturals and the convergence behaviour of sums related to it; is there a theory that formally links these topics? For example, the sum of reciprocals of 1 / n^{1 + e} converges if e > 0, does this entail that, in some sense, for almost all n and given e > 0 that there is a prime in the interval [n^{1 + e}, (1 + n)^{1 + e}]; or that, probabilistically speaking, it is "likely" that there is a prime in this interval? What about cases where series A converges to a, B to b, and a > b; can we say something about the expected ratio of A-elements to B-elements in a given interval (assuming A and B are somewhat "naturally" distributed)? I'd be interested in any results of this type, particularly if they are part of a larger methodology of generating such links. I'm not specifically interested in primes as much as the general idea, but it seems like the study of primes would be ripe with such results given it's nature. Thank you for any, and all, help:-)Phoenixia1177 (talk) 23:25, 8 June 2013 (UTC)[reply]

Such sequences are sometimes called small sets. The Muntz-Szasz theorem came to mind when I first read your question, for what it's worth. I attended a lecture a few years back by Per Enflo on some generalizations. I may have the notes somewhere. Sławomir Biały (talk) 23:53, 8 June 2013 (UTC)[reply]

Dirichlet density (often more tractable than natural density) and Abstract analytic number theory may be of interest too. Noticed that nearby, unfortunately, Probabilistic number theory is a stub.John Z (talk) 13:18, 10 June 2013 (UTC)[reply]

Thank you both:-) The articles linked to are fascinating, as are the articles they link to. The Muntz-Szasz theorem is extremely interesting, I never would have thought to look into that connection. This brings me to two further questions/comments. The open problem posed by Erdos in the small set article appears to have some obvious links to combinatorics on words, specifically irrationality and how subwords repeat (link the alphabet up with lengths of arithmetic progressions...); at any rate, is there any literature that links these subjects, interpret as broadly as you'd like? And, while I'm sure there are lots of journal articles to look at, I don't have access to journals, are there any textbooks that cover density and it's relation to integer sequences? Or, relate integer sequences to analytic techniques; but not analytic number theory, something more related to density and combinatorics? Sorry if this is a lot of follow up, for some reason this subject is fascinating to me, but sadly I don't know how to get started studying it with my resources. Thank you for all, and any further, help:-)Phoenixia1177 (talk) 07:14, 11 June 2013 (UTC)[reply]