Diophantine Frobenius Problems — counting theory, generating function and zeta functions

When a_1, ...,a_m are relatively prime positive integers, the number of solutions of the linear Diophantine equation a_1 x_1 + … + a_m x_m = b in non-negative integers x_1, ...,x_m, for any integer b, is our concern. We show several formulas to give the largest integer b without solution. Then we discuss the generating function of the number of solutions. Finally, we derive an explicit expression for an inverse power series over the gaps values of numerical semigroups generated by two integers. It implies a set of new identities for the Hurwitz zeta function.