Characterizations of Lambek-carlitz Type

We give Lambek-Carlitz type characterization for completely multiplicative reduced incidence functions in Möbius categories of full bi-nomial type. The q-analog of the Lambek-Carlitz type characterization of exponential series is also established. 1. An arithmetical function f is called multiplicative if (1.1) f (mn) = f (m)f (n) whenever (m, n) = 1 and it is called completely multiplicative if (1.2) f (mn) = f (m)f (n) for all m and n. Lambek [5] proved that the arithmetical function f is completely multiplicative if and only if it distributes over every Dirichlet product: (1.3) f (g * D h) = f g * D f h , for all arithmetical functions g and h. g * D h is defined by: (g * D h)(n) = d|n g(d)h n d. Problems of Carlitz [1] and Sivaramakrishnan [12] concern the equivalence between the complete multiplicativity of the function f and the way it distributes over certain particular Dirichlet products. For example, Carlitz's Problem E 2268 [1] asks us to show that f is completely multiplicative if and only if (1.4) f (n)τ (n) = d|n f (d)f n d (∀n ∈ N *) , that is if and only if f distributes over ζ * D ζ = τ , where ζ(n) = 1, ∀n ∈ N * , and τ (n) is the number of positive divisors of n ∈ N *. 2. Möbius categories were introduced in [7] to provide a unified setting for Möbius inversion. We refer the reader to [2] and [8] for the definitions of a Möbius category and of a Möbius category of full binomial type, respectively. In the