Harmonic Numbers over Additive and Multiplicative Semigroups

In this work, we investigate the integrality and divisibility problems of harmonic numbers over additive and multiplicative semigroups. We first study the $S$-harmonic numbers $h_S(n)$, which we define as the sum of the reciprocals of non-zero elements in a semigroup $S$ up to $n$. We prove that $h_S(n)$ is never an integer for all sufficiently large integers $n$ (Theorem A), and we explicitly show that the lower bound for this non-integrality depends on the algebraic invariants of the semigroup. Secondly, we introduce the smooth harmonic numbers $H(n; z, A)$, which are weighted sums of reciprocals of positive integers where the weights possess a smoothness property. We establish a non-integrality result for these sums under a density condition on the smoothness function $z(n)$ and extend this framework to the number fields (Theorem B). In addition to these two main theorems, we analyze the arithmetic behavior of the Beurling harmonic numbers $H(n, \mathcal{P})$, and examine the finiteness of their divisibility sets $J(p, \mathcal{P})$. Finally, we explore the integrality of the Beurling hyperharmonic numbers $H(n, r, \mathcal{P})$.