Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define $$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$ for all real numbers $x \in (M(d), \infty)$. Here $M(d)$ is chosen large enough so that all the iterated logs are positive real, i.e., $M(0) = 0$ and $M(1) = 1$, and $M(d) = e^{M(d-1)}$ for $d > 1$.
Let $S$ be a subset of ${\mathbb N}$. Let's say that $S$ is $\lambda$-analytic if there exists a finite sequence $t$ as above such that the limit $$L = \lim_{X \rightarrow \infty} \frac{\# (S \cap \{0,1,\ldots, X\})}{\lambda^t(X)}.$$ exists and $0 < L < \infty$. In such a case, we say $S$ is $\lambda$-analytic with exponent $t$.
Many important theorems and conjectures in analytic number theory can be rephrased in the form "This neat subset of ${\mathbb N}$ is $\lambda$-analytic with exponent $t$." E.g., the prime number theorem says that the set of primes is $\lambda$-analytic with exponent $(1,-1)$. For another example, the set $\{ x^2 + y^2 : x,y \in {\mathbb N} \}$ is $\lambda$-analytic with exponent $(1,-1/2)$.
I'm wondering, broadly speaking, which sets of natural numbers might be the subject of such theorems and conjectures. I mean, number theorists care about primes, sums of two primes, numbers of the form $x^2 + y^4$, etc. What makes some sets of natural numbers $\lambda$-analytic... even if we can't yet prove the exponent? This seems like a crucial sort of meta-question in analytic number theory.
For example, are all recursive subsets of ${\mathbb N}$ expected to be $\lambda$-analytic?
Is there some other complexity class for subsets of ${\mathbb N}$, where everything is expected to be $\lambda$-analytic?
If $A$ and $B$ are $\lambda$-analytic subsets of ${\mathbb N}$, is their sumset $A+B$ always $\lambda$-analytic?
References appreciated! I don't know if this idea of $\lambda$-analytic subsets of ${\mathbb N}$ is in the literature somewhere, under a different name.
