Density and Sums of Reciprocals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:08:20Z http://mathoverflow.net/feeds/question/32988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32988/density-and-sums-of-reciprocals Density and Sums of Reciprocals Nathan St. John 2010-07-22T18:41:33Z 2010-07-22T19:01:42Z <p>Is there a notion of density for a (strictly increasing) sequence of natural numbers that decides whether the sum of the reciprocals of that sequence converges?</p> http://mathoverflow.net/questions/32988/density-and-sums-of-reciprocals/32991#32991 Answer by J. H. S. for Density and Sums of Reciprocals J. H. S. 2010-07-22T18:53:01Z 2010-07-22T19:01:42Z <p>The notion of natural density gives a suficient condition. Namely, one can prove that if $A \subseteq \mathbb{N}$ has positive upper n. density then $\sum_{a \in A} \frac{1}{a}$ diverges. This condition is not a necessary one, though. A well-known result in Number Theory ascertains that the series of the reciprocals of the prime numbers diverges whereas $\pi(n) = o(n)$.</p> <p>In the following list you are to encounter some previous discussions on MO that are closely related to the current inquiry of yours:</p> <p>[<strong>1</strong>] <a href="http://mathoverflow.net/questions/13230/erdos-conjecture-on-arithmetic-progressions" rel="nofollow">http://mathoverflow.net/questions/13230/erdos-conjecture-on-arithmetic-progressions</a></p> <p>[<strong>2</strong>] <a href="http://mathoverflow.net/questions/4596/on-the-series-1-2-1-3-1-5-1-7-1-11" rel="nofollow">http://mathoverflow.net/questions/4596/on-the-series-1-2-1-3-1-5-1-7-1-11</a></p> <p><strong>References</strong></p> <p><strong>I</strong>. Erdös's brilliant proof of the divergence of $\sum_{p} \frac{1}{p}$: <a href="http://www.renyi.hu/~p_erdos/1938-13.pdf" rel="nofollow">http://www.renyi.hu/~p_erdos/1938-13.pdf</a></p>