Natural generalization of Euler-type constants
问题内容
Let $$ \gamma=\lim_{x\to\infty}\left(\sum_{n=1}^x \frac1n-\log x\right) = 0.57721... $$ be Euler’s constant, and let $$ M=\lim_{x\to\infty}\left(\sum_{p \text{ prime}}^{p\le x}\frac1p-\log\log x\right)=0.26149... $$ be Mertens’ constant. These are two examples of reciprocal sums with (iterated-) logarithmic growth: \begin{align*} \sum_{n\le x}\frac1n = \log x+\gamma+o(1),\ \sum_{p\le x}\frac1p = \log\log x+M+o(1). \end{align*} Is there a natural continuation of this pattern. More precisely, does there exist a nested family of sets $ S_0 \supset S_1 \supset S_2 \supset \cdots $ with $ S_0=\mathbb N, S_1=\mathbb P, $ together with a sequence of constants $(C_0,C_1,C_2,\cdots)$ with $C_0=\gamma, C_1=M, $ such that for each $k\ge 0$: \begin{align*} \sum_{n\in S_k}^{n\le x}\frac1n=\log^{[k+1]}(x)+C_k+o(1), \end{align*}
where $\log^{[k]}$ denotes the $k$-fold iterated logarithm?
Can one define $S_k$ for $k\ge 2$ in a natural arithmetic way, rather than by an artificial construction tuned to force the asymptotic. If so, what can be said about the $S_k$ and $C_k$?
回答 (1)
I am not a number theorist, so I don't know any "natural" examples of such a sequence $S_k$. But if you're willing to use arithmetic notions to (somewhat arbitrarily) cook up such an example, then here is an attempt.
Quantities like $\log^{[k]}x$ appear naturally as asymptotics to $$\sum_{n\leq x}\frac1n \prod_{m=0}^{k-1}\frac{1}{\log^{[m]}n},$$ of which your two examples fall into. You can view the derivation of the $\ln\ln x$ bound for sums of prime reciprocals probabilistically: the probability that a given $n$ is prime is $1/\log n$ by the prime number theorem, so $$\sum_{\substack{p\leq x\\ p \text{ prime}}} \frac1p \sim \sum_{n\leq x} \frac1{n\log n} \sim \log\log x.$$ To introduce more iterated logs in the denominator of the sum, $\log n$ counts the number of digits of $n$, so the probability that a given number is prime and also has a prime number of digits is $$\frac{1}{\log n} \cdot \frac{1}{\log\log n}.$$ Which means that, for $D(p)$ the number of digits of $p$, $$\sum_{\substack{p\leq x\\ p, D(p) \text{ prime}}} \frac1p \sim \sum_{n\leq x} \frac1{n\log n \log\log n} \sim \log\log\log x.$$ Similarly, the probability that a given number is prime, has a prime number of digits, and the number of digits of its number of digits is prime is $$\frac{1}{\log n} \cdot \frac{1}{\log\log n}\cdot\frac{1}{\log\log\log n},$$ and one can evaluate the sum to get a $\log\log\log\log x$ asymptotic. Therefore, continuing this pattern we see that $$\sum_{\substack{p\leq x\\ p, D(p), \dots, D^{k-2}(p) \text{ prime}}} \frac1p \sim \log^{[k]}x.$$ Actually determining properties of the constants in this setting, or even that such constants exist, seems to be difficult.