\(p\)-adic valuation of harmonic sums and their connections with Wolstenholme primes (Q6572376)

In the literature, many properties of the harmonic numbers have been discovered, and many scientists have studied their applications using different methods.\N\NThe \(n\)-th harmonic number is defined as the partial sum of the well-known harmonic series \(H(n) := 1 + 1/2 + \dots + 1/n\). It has been known since the nineteenth century that by the Wolstenholme Theorem, for any prime \(p\geq5\) the numerator of \(H(p-1)\) is multiple of \(p^2\). In the last century, many researchers have investigated the arithmetic properties of harmonic numbers and related problems.\N\NIn this paper, the authors explore a conjecture posed by \textit{A. Eswarathasan} and \textit{E. Levine} [Discrete Math. 91, No. 3, 249--257 (1991; Zbl 0764.11018)] on the distribution of \(p\)-adic valuations of harmonic numbers that states that the set \(J_p\) of the positive integers \(n\) such that \(p\) divides the numerator of \(H(n)\) is finite. They prove two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the \(p\)-adic valuation of \(H(p^mn)\) when the \(p\)-adic valuation of \(H(n)\) equals exactly \(3\).