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For a positive integer $n$, write $[n]=\{1,\ldots,n\}$ and put

$$R_n=\#\left\{S\subseteq[n]:\sum_{s\in S}\frac1s\le1\right\}, \qquad E_n=\#\left\{S\subseteq[n]:\sum_{s\in S}\frac1s=1\right\}.$$

These are subsets, so denominators are distinct. The empty subset is counted by $R_n$ and not by $E_n$. In particular, $E_n\le R_n$.

Let $H_0=0$ and $H_n=\sum_{i=1}^n1/i$. On the finite uniform probability space $\{-1,1\}^n$, let $\varepsilon_1,\ldots,\varepsilon_n$ be the coordinate signs and set

$$Z_n=\sum_{i=1}^n\frac{\varepsilon_i}{i}, \qquad V_n=\sum_{i=1}^n\frac1{i^2}.$$

The signs are independent, each with probabilities $1/2,1/2$. Every sign vector corresponds to exactly one subset through $\mathbf1_{i\in S}=(1+\varepsilon_i)/2$. Reflection of all signs preserves the uniform measure. It equates the probabilities of the lower and upper tail events used in the signed reformulation; it does not identify those events point by point.

All logarithms in the proofs are natural. We use $\cosh y=(e^y+e^{-y})/2$. An eventual assertion means that there is an integer $n_0$ such that it holds for every integer $n\ge n_0$. No explicit value of $n_0$ is asserted in the main theorem.

Source. Steinerberger, arXiv:2403.17041v5, 28 April 2024, pp. 1–2. The symbols $R_n,E_n,Z_n,V_n$ are convenient names used by this compilation.

Bears on. #297 only by fixing the counts and notation that the Theorem and the other result pages of this source use; $E_n$ is the problem's count.