Harmonic Subseries and a Problem

I’ve spent a lot of my time recently thinking about harmonic subseries and whether or not they converge. I came across the paper Summing Curious, Slowly Converging, Harmonic Subseries and I thought it contained a result tasty enough to share on my blog.

Remember the harmonic series:

\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots

This is often the first example students of maths will see of an infinite sum whose summands get closer to zero yet the sum still diverges. Schmelzer and Baillie prove that if you remove every term in the above series whose denominator contains some fixed string of numbers it will then converge!

For example, if you chose the string “235” and proceeded to delete every term in the series which had a 235 in its denominator (e.g. 1/123576) then you would have a convergent sum my friend. What a neat statement!

Diverging (geddit?) from this result for a second, let’s consider the fact that the sum of the reciprocals of the primes diverges, that is

\sum_{p} \frac{1}{p} = \infty.

The divergence of the reciprocal prime series coupled with this result can tell you a few little things about the distribution of the primes, the most obvious of which being that there are infinitely many primes whose digits contain any fixed string, for deleting such a set of primes would render the sum finite. For example, there are infinitely many primes which contain the string 666. Also, there are infinitely many prime numbers which contain your mobile number, your credit card details and so on. The sequence of primes is rather intrusive!

While we are on the hot topic of prime distributions, I want to talk a bit about a thought I had the other day. Most people know about the twin prime conjecture, an unproved statement which says that there are infinitely many prime numbers which differ by 2. The sequence of these twin primes begins as follows

(3,5), (5,7), (11,13), (17,19), \dots

and it is not known as to whether or not it continues forever.

So here I was, sitting around (story of my life) and trying to think up some sort of connection between the divergence of \sum 1/p and the twin prime conjecture. In particular, I thought that if the gap of length 2 persisted throughout the sequence of prime numbers, this might be the reason they “grow so slowly”, in the sense that their reciprocal sum diverges. From this, I conjectured (sounds grand, right?) the following:

Let \sum 1/a_n be a harmonic subseries with the property that for any positive integer G, there exists another positive integer N such that n>N implies that a_{n+1} - a_n > G. Then \sum 1/a_n converges.

Basically what I was thinking was that if you had a harmonic subseries where any finite gap in the denominators ceased after some time, then the sum would ultimately converge. This would be great because the contrapositive says that divergence of a harmonic subseries implies at least one infinitely repeated gap in the sequence of denominators. If this were true, what a result that would be in the context of prime numbers!

Unfortunately, I quickly devised a counterexample. Can you find one?

5 thoughts on “Harmonic Subseries and a Problem

  1. Ok I think I’ve a counter-example. Take the sequence A=(1/3+1/4)+1/2*(1/5+1/6+1/7+1/8)+1/3*(1/9+…+1/16)+… so a_n has arbitrary large gaps for n big enough. Then A > 2*1/4 + 4*1/2*1/8 + 8*1/3*1/16+…= \sum 2^n/n*2^(n+1) = \sum 1/2n = \infty

    1. Cheers Tom! Personally, I think the statement of the result warrants a place in first year calculus courses. I’ve used it in a few talks on primes, just to say something like “there are infinitely many primes which contain your credit card number” or something similiar. Indeed, the result is neat!

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