Introduction:


Let A be a subset of the naturals such that $\sum\_{n\in A}\frac{1}{n}=\infty$. The [Erdos Conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions) states that A must have arithmetic progressions of arbitrary length. 


Question:


I was wondering how one might go about *categorizing* or *generating* the divergent series of the form in the introduction above. I'm interested in some particular techniques and I list some examples below:


If we let $S$ be the set of such divergent series: $S=\left[ A: \sum\_{n\in A}\frac{1}{n}=\infty, \ A\in\mathbb{N} \right]$, what kind of operations are there that would make S a group, or at the very least a semigroup? I'm rather vague on what the operatons should be for a reason, because although I presume trivial operations exist, their usefulness in understanding the members of $S$ would be questionable. 


Alternately, can one look at these divergent sums through the technique of Ramanujan summation (think: $1+2+3+\ldots =^R -\frac{1}{12}$, $R$ emphasizing Ramanujan summation)? The generalizations of Ramanujan summation (a good reference  [here](http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf) ) allow one to assign values to some of these series and give some measure of what kind of divergence is occurring. Moreover, basic series manipulations that hold for convergent series tend to carry over to Ramanujan summation, so can one perhaps look at the set $S$ above as a set of equivalence classes in the sense of two elements being equivalent if they share the same Ramanujan summation constant. 


Thanks in advance for any input!



Erdos Conjecture on arithmetic progressions
Most of the point of this answer is to promote a piece of terminology:


Three years ago I first taught a number theory course at UGA in which I made the following definition: a subset $A$ of the positive integers is **substantial** if


$\sum\_{n \in A} \frac{1}{n} = \infty$.


A little bit of discussion of this concept occurs in Section 4 of


[http://alpha.math.uga.edu/~pete/4400primes.pdf](http://alpha.math.uga.edu/%7Epete/4400primes.pdf)


I do mention the Erdos(-Turan?) conjecture about arithmetic progressions in substantial sets. For the purpose of constructing examples, possibly the remark I make about any set with positive upper density being substantial will be of most use to you.


Those notes don't contain a proof of that, but a proof of this and more can be found in a very (very!) nice final project done in this class by (then) undergraduate Alex Rice. Sadly he never gave me an electronic copy in a form that I was able to upload to my webpage. If you want to see his writeup, let me know and I'll bug him about this again: the winds of fate have blown him around a bit, but he is now again a UGA (graduate) student taking a number theory course from me.


Finally, to answer one of your questions in a cheap way: yes, the set of substantial subsets of $\mathbb{Z}^+$ certainly forms a semigroup under union. This seems like a completely unhelpful observation, but who knows...


