For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by \begin{align*} \underline{d}(A)=\liminf_{N\to\infty} \frac{A\cap [1,N]}{N},\\ \bar{d}(A)=\limsup_{N\to\infty} \frac{A\cap [1,N]}{N},\\ \underline{\delta}(A)=\liminf_{N\to\infty} \frac{1}{\ln N}\sum_{{n\in A\cap [1,N]}}\frac{1}{n},\\ \bar{\delta}(A)=\limsup_{N\to\infty} \frac{1}{\ln N}\sum_{{n\in A\cap [1,N]}}\frac{1}{n}. \end{align*} It is known that for each $A\subseteq \mathbb{N}$ we have $$ \underline{d}(A) \leq\underline{\delta}(A)\leq\bar{\delta}(A)\leq\bar{d}(A). $$ Asymptotic densities have natural analogs for $A\subseteq\mathbb{N}^d$ ($d\ge 1$) given by \begin{align*} \underline{d}(A)=\liminf_{N\to\infty} \frac{A\cap [1,N]^d}{N^d},\\ \bar{d}(A)=\limsup_{N\to\infty} \frac{A\cap [1,N]^d}{N^d}. \end{align*} Have analogs of upper/lower logarithmic densities been defined/studied? In $\mathbb{N}^d$ or $\mathbb{Z}^d$? How about more general setting (countable amenable groups)?
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$\begingroup$ What about iterated logarithmic density, for example $\liminf \dfrac{1}{\log\log N}\sum_{p\in A\cap [2, p_{\pi(N)}]}\dfrac{1}{p}$ when $N\to\infty$? $\endgroup$ – Sylvain JULIEN Jun 29 '16 at 19:40

$\begingroup$ I'm not sure what's been defined, but maybe a natural averaging scheme would be to sum with weights $1/\n\^d$, where $d$ is the dimension. The rationale for this is that the weights, summed over an $n$ball are of order $\log n$. $\endgroup$ – Anthony Quas Jun 30 '16 at 0:23
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The natural $k$dimensional analogue of logarithmic density is $$ \lim_{x \rightarrow \infty} \frac{1}{(\log x)^{k}} \sum_{\substack{n_1, \ldots, n_k \leq x \\ (n_1, \ldots, n_k) \in S}} \prod_{i = 1}^{k} \frac{1}{n_i} $$ and same for the variant with $\limsup$ and $\liminf$. I'm sure people used this when it was convenient.