  The multivariate arithmetic Tutte polynomial of arithmetic matroids is a
generalization of the multivariate Tutte polynomial of matroids. In this note,
we give the convolution formulas for the multivariate arithmetic Tutte
polynomial of the product of two arithmetic matroids. In particular, the
convolution formulas for the multivariate arithmetic Tutte polynomial of an
arithmetic matroid are obtained. Applying our results, several known
convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1,
Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs
presented here are significantly shorter than the previous ones. In addition,
we obtain a convolution formula for the characteristic polynomial of an
arithmetic matroid.
