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It seems natural to consider $\lim_{q \rightarrow 1^-} \sum_{n \in S} q^n - \sum_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same density; for instance, $\{0,2,4,...\}$ and $\{1,3,5,...\}$ both have density 1/2, but the first set might be said to contain "half an element more" than the second, based on the fact that $1-q+q^2-q^3+...$ converges to 1/2 as $q$ goes to 1. Has this (non-Archimedean, non-translation-invariant) refinement of the concept of density, or anything similar to it, been developed anywhere?

Put $|S|_q = \sum_{n \in S} q^n$. Just as $|\{0,2,4,6,...\}|_q-|\{1,3,5,7,...\}|_q \rightarrow 1/2$ as $q \rightarrow 1^-$, it seems empirically that $$|\{0,1,4,9,16,...\}|_q-|\{0,2,6,12,20,...\}|_q \rightarrow 1/2,$$ $$|\{0,1,5,12,22,...\}|_q-|\{0,2,7,15,26,...\}|_q \rightarrow 1/3,$$ and $$|\{0,1,3,6,10,..,\}|_q - \sqrt{2} |\{0,1,4,9,16,...\}|_q \rightarrow \sqrt{2}/2.$$ Is there a place in the literature where results like this can be found, as part of a general setup for measuring sets that "sees" things at a finer level than mere density? To give one last example, racing "evil" versus "odious" numbers, it is easy to prove that $$|\{0,3,5,6,9,10,12,15,...\}|_q-|\{1,2,4,7,8,11,13,14,...\}|_q \rightarrow 0.$$

[ADDED ON MAY 17, 2017: There is now (at least) one article that uses analytic tools to generalize the notion of cardinality of sets of natural numbers: my preprint "One-Dimensional Packing: Maximality Implies Rationality", available at https://arxiv.org/abs/1704.08785 ]

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  • $\begingroup$ Sorry but if you are going to base your article on this, it would be a clear case of plagiarism. You took the idea expressed here: mathoverflow.net/questions/215762/… and changed Ramanujan summation to Abel summation. If you are still going to make this article I at least encourage you to cite the original post (another my post has been cited by Joel Hamkins in a paper so this should not be a problem). $\endgroup$ – Anixx Jan 27 '17 at 1:52
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    $\begingroup$ Ilya's presumption that I "took the idea expressed here: mathoverflow.net/questions/215762/… and changed Ramanujan summation to Abel summation" is incorrect; I can say more about the thread of rumination that led me to Abel summation (see for instance my 2013 MO post "mathoverflow.net/questions/123411/…) but I'd rather not clutter this thread with those details. I wasn't influenced by Ilya's work, but I agree that there are parallels, and I will be happy to cite Ilya's work when my work leads to an article. $\endgroup$ – James Propp Jan 28 '17 at 4:29
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Different notions of summability have indeed been considered before (see in particular Abel summation). For example, in the study of primes a famous observation of Chebyshev is that there are more primes of the form $3\pmod 4$ than $1\pmod 4$. Usually this is interpreted in the sense of the number of primes up to $x$ that are $3\pmod 4$ is more than the number of primes that are $1\pmod 4$; this is true most of the time, but not always. Chebyshev's original formulation was along the lines of your question: he wanted to know if $$ \lim_{q \to 1^-} \Big( \sum_{p \equiv 3\pmod 4} q^{p} - \sum_{p\equiv 1\pmod 4} q^p\Big) = \infty. $$ Hardy and Littlewood and Landau already noted a hundred years back that this is equivalent to the Riemann hypothesis for $L(s,\chi_{-4}) =1/1^s-1/3^s+1/5^s-\ldots$. The same question for natural density is more subtle, involving not just RH but also relations among the zeros. In general the smooth sums in the question would be easier to handle than corresponding problems for the counting functions of the sets. For some references on the particular example given here, see Ford and Konyagin, Rubinstein and Sarnak, or Granville and Martin.

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  • $\begingroup$ Put $|S|_s = \sum_{n \in S} n^{-s}$ (where $S$ is a set of positive integers). Then Lucia is pointing out that the behavior of $|S|_s - |T|_s$ as $s \rightarrow 0$ is also a natural way to measure the difference in size between $S$ and $T$. When $S=\{1,3,5,\dots\}$ and $T=\{2,4,6,\dots\}$, one has $|S|_s - |T|_s \rightarrow 1/2$ as $s \rightarrow 0$ (see en.wikipedia.org/wiki/Dirichlet_eta_function); I would be interested in knowing what zeta-regularization yields for the other pairs of sets described in the original post. $\endgroup$ – James Propp Sep 5 '16 at 17:59
  • $\begingroup$ I don't think I'm pointing out the zeta function regularization (although of course one could study that too) -- in my answer I was describing just the Abel regularization of the original question. $\endgroup$ – Lucia Sep 5 '16 at 18:07
  • $\begingroup$ Lucia is correct; I mis-characterized her reply. In any case, my search for information is fairly broad, so other regularization schemes would interest me as well (especially theorems that assert equality between the answers obtained from different regularization procedures). $\endgroup$ – James Propp Sep 5 '16 at 18:14
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Your post expresses the same idea about comparison of infinite sets as in my previous post with the difference being that you use Abel summation while I did use Ramanujan's summation and Zeta regularization. To be more precise, I used Faulhaber's formula for summation because it gives the same results as Ramanujan's summation (I do not know a proper term for this summation method):

$$\sum _{x\ge0}^\Re f(x)=-\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(0)$$

Now, addressing your concerns, expressed in the comments to the other answer, there is a set of similar summation methods that are mostly compatible with each other. One can see that some of them depend only on the values of the series at the integer points while others involve integrals and derivatives. So, it clearly means that there can be constructed examples where these methods give different results.

My strong conviction, even if I do not have a proof, is that all these methods should give the same results for "well-behaving" functions. By "well-behaving" I mean functions that are equal to their Newton series:

$$f(x) = \sum_{k=0}^\infty \binom{x-a}k \Delta^k f\left (a\right)$$

I call such functions "discrete-analytic", "Newton-analytic" or "Newtonian".

The criterion can also be written in the following form:

$$f(x)=\lim_{n\to\infty}\frac{\sum_{k=0}^{n} \frac{(-1)^k f(k)}{(x-k)k!(n-k)!}}{\sum_{k=0}^{n} \frac{(-1)^k }{(x-k) k!(n-k)!}}$$

Some functions would have this series divergent, but their two-directional expansion would converge:

$$f(x)=\lim_{n\to\infty}\frac{\sum _{k=-n}^n \frac{(-1)^k f(k)}{(x-k) (k+n)! (n-k)!}}{\sum _{k=-n}^n \frac{(-1)^k}{(x-k) (k+n)! (n-k)!}}$$

I would include such functions into "Newtonian" as well.

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