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$R_n \geq n/16$ can be obtained by starting from an arbitrary $f$ and then switching pairs of transpositions to get rid of any overlapping pairs whose images are too close to each other. Suppose $r(f …

This formula can actually be proved using only properties of the Gamma function already known to Gauss, with no need to invoke special values of Dirichlet series. The relevant identities are $$ \Gamm …

Does this reduce to values of a known special function for arbitrary real (or complex) $s$? Answered by Johannes Trost in a comment: it's also known as a "Roman harmonic number". But this …

Here's how to average $e^{sx}$ over $A_t$. Let $r = t/2$, so at the $N$-th step of the construction of $A_t$ we have the disjoint union $A_t^{(N)}$ of $2^N$ intervals of length $r^N$ whose left endpo …

Not in general. An explicit and elementary counterexample is the sparse triangular matrix with $1$'s on the diagonal and $-1$'s just above it: the inverse is the triangular matrix with every entry …