Some Variations on the Dedekind Conjecture

In this note we prove a group theoretic statement about expressing certain characters of a finite solvable group as a sum of monomial characters. This is used to prove holomorphy of certain products of Artin L-functions which can be thought of as a variant of the Dedekind Conjecture. This variant is then used to improve, in the solvable case, a certain inequality due to R. Foote and K. Murty which bounds the orders of some Artin L-functions, at an arbitrary but fixed point in the complex plane, in terms of the order of a suitable quotient of Dedekind zeta functions. This improved inequality has a rather striking consequence regarding non-existence of simple zeros or simple poles in such quotients.