Journal of the Ramanujan Mathematical Society

A. Raghuram ¹, Naomi Tanabe ¹

Affiliations
1 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK-74078, US

Abstract

Shimura proved the following fundamental result (see [37, Theorem 4.3]) on the critical values of the standard L-function attached to a holomorphic Hilbert modular form.

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M. Ram Murty ¹, A. Raghuram ²

Affiliations
1 Department of Mathematics, Queen's University, Kingston, K7L 3N6 Ontario, CA
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, IN

Abstract

In this paper 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.

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