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Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture.

Let Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer n > = 2. In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers. [ABSTRACT FROM AUTHOR]

Abstract Text Let v n be the denominator of the n -th harmonic number H n = 1 + 2 − 1 + ⋯ + n − 1. In this paper, we prove the following unusual result: the set of positive integers n with v n = v n + 1 has density one. We also prove that the same result is true for the denominators of alternating harmonic numbers. Video For a video summary of this paper, please visit https://youtu.be/Sk0zMSKPe3M. [ABSTRACT FROM AUTHOR]

Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]