New Connections Between Functions from Additive and Multiplicative Number Theory - Merca, Mircea (Mediterranean journal of mathematics, 2018)
In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and q -series methods. These general results are illustrated considering relationships between the gcd-sum function and integer partitions. Even if there is in the literature a large number of works in which many properties of the gcd-sum function are studied, connections between the gcd-sum function and integer partitions have not been remarked so far. Our general results can be used to provide new connections between the partitions and many classical special arithmetic functions often studied in multiplicative number theory: the Möbius function μ ( n ) , Euler’s totient φ ( n ) , Jordan’s totient J k ( n ) , Liouville’s function λ ( n ) , the von Mangoldt function Λ ( n ) , the divisor function σ x ( n ) , and others.