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 hS(n), which we define as the sum of the reciprocals of non-zero elements in a semigroup S up to n. We prove that hS(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, P), and examine the finiteness of their divisibility sets J(p, P). Finally, we explore the integrality of the Beurling hyperharmonic numbers H(n, r, P).