On the \(2\)-adic valuation of differences of harmonic numbers (Q6542782)

After having defined the harmonic numbers \(H_{n} = \sum_{k=1}^{n} 1 / k \) (with \(H_0 = 0\)), their differences \(H_{n} - H_{m} = \sum_{k=m+1}^{n} 1 / k \) (with \(0 \leq m \leq n-1\)), and the base \(2\) expansion of \(n\) as \(n=\sum_{i=1}^{t} 2^{a_{i}}\) (being \(a_{i} \in \mathbb{N}\)), the author recalls from \textit {K. Conrad} [``The \(p\)-adic growth of harmonic sums'', Preprint, \url{https://kconrad.math.uconn.edu/blurbs/gradnumthy/padicharmonicsum.pdf}] the exact \(2\)-adic valuation \N\[\N\nu_{2}\left(H_{2 n}-H_{n}\right)=-\left\lfloor\log _{2}(2 n)\right\rfloor, \quad n \geq 1,\N\]\Nin order to prove that, if \(M_{j}=\sum_{i=t-j+1}^{t} 2^{a_{i}}\) (with \(j=1,2, \ldots, t\)) and \(M_{0}=0\), then \N\[\N\nu_{2}\left(H_{n}-H_{m}\right)=-a_{t-j+1}, \quad M_{j-1} \leq m < M_{j}, \N\]\Nand further that, if \(n=2^{a_{t}}\) (with \(a_{t} \geq 0\)) is a power of \(2\), then \N\[\N\nu_{2}\left(H_{n}-H_{n-m}\right)=-a_{t}, \quad 1 \leq m \leq n,\N\]\Nand finally that \N\[\N\nu_{2}\left(\prod_{m=0}^{n-1}\left(H_{n}-H_{m}\right)\right)=-\sum_{i=1}^{t} a_{i} 2^{a_{i}}.\N\]\NHence the paper focuses on the integer sequence \(a(n)=\sum_{i=1}^{t} a_{i} 2^{a_{i}}\) given by the weighted digit sum of \(n\) in base \(2\), proving via induction that, for \(n \geq 1\), \N\[\N(n+1) \log _{2}(n+1)-2 n \leq a(n) \leq n \log _{2} n,\N\]\Nand finding, subsequently, the sharp bounds for \(\sum_{m=0}^{n-1} \nu_{2}\left(H_{n}-\right.\) \(\left.H_{m}\right) / n\) (i.e., the average of the \(2\)-adic orders of the differences): \N\[\N-\log _{2}(n) \leq \frac{1}{n} \sum_{m=0}^{n-1} \nu_{2}\left(H_{n}-H_{m}\right) \leq-\frac{n+1}{n} \log _{2}(n+1)+2.\N\]\NEventually, the paper establishes auxiliary results such as lower bounds on the \(2\)-adic valuations of elementary symmetric functions related to the OEIS sequence \texttt{A061168} and the \(2\)-adic properties of a lacunary binomial sum that the author had partially studied in [\(p\)-Adic Numbers Ultrametric Anal. Appl. 15, No. 1, 23--47 (2023; Zbl 1520.11030)].