The Smith arithmetic determinant is investigated in this paper. By using two different methods, we derive the explicit formula for the Smith arithmetic determinant. Keywords—Elementary row transformation, Euler function, Matrix decomposition, Smith arithmetic determinant.

Let V (n) denote the number of positive regular integers (mod n) that are ≤ n, and let Vk(n) be a multidimensional generalization of the arithmetic function V (n). We find the Dirichlet series of Vk(n) and give the extremal orders of some totients involving arithmetic functions which generalize the sum-of-divisors function and the Dedekind function. We also give the extremal orders of other tot...

Zeta values in Tate algebras were introduced by Pellarin 2012. They are generalizations of Carlitz zeta and play an increasingly important role function field arithmetic. In this paper we prove a conjecture on identities for these values. The proof is based arithmetic properties explicit formula Bernoulli-type polynomials attached to algebras.