This article explores the deep connections between arithmetic functions and integer partitions through the application of the Fine and Jameson-Schneider theorems. By employing Bell polynomials, several classical arithmetic functions, including the divisor function, sum of divisors, Euler’s totient function, Möbius function, and sums of two squares, are represented in terms of integer partitions. The study highlights how combinatorial structures provide alternative approaches to evaluate and interpret number-theoretic functions. Furthermore, recurrence relations and identities are established, enriching the theoretical framework linking partition theory with analytic number theory. These results contribute to a broader understanding of arithmetic properties and their combinatorial representations, offering potential applications in both pure mathematics and related computational fields.