Riemann Hypothesis Analogue for the Invariant Ring ℂ[𝑥,𝑦] Dn Notes on ℂ[𝑥,𝑦] D8p
E. Nocon (pp.53-73)
Abstract
The Riemann hypothesis is one of the most talked about problems in number theory. It is usually expressed in terms of the Riemann zeta function having zeros that describe the distribution of prime numbers. Then, for years, Riemann hypothesis analogues (RHAs) came to the attention of mathematicians. There were studies about global zeta functions of function fields (Artin, 1924; Weil, 1949). Then, in the past years, applications of RHA caught the attention of coding theorists (Duursma, 2003; Harada & Tagami, 2007; Kim & Hyun, 2012). In this paper, we consider an RHA for the invariant ring ℂ[𝑥,𝑦] Dn, where 𝐷𝑛 is the dihedral group of order 2𝒏. We determine whether the Riemann hypothesis holds for all the extremal polynomials in ℂ[𝑥,𝑦] Dn. Results obtained for the case D8 are used to examine Type I codes and their corresponding zeta functions.