An abundance of invariant polynomials satisfying the Riemann hypothesis

TL;DR

This paper constructs numerous invariant polynomials satisfying the Riemann hypothesis, extending the concept beyond existing codes and providing explicit examples and theoretical tools like an analogue of the Eneström-Kakeya theorem.

Contribution

It demonstrates the abundance of invariant polynomials satisfying the Riemann hypothesis through explicit construction and theoretical analysis.

Findings

Many invariant polynomials satisfy the Riemann hypothesis.

Explicit constructions of such polynomials are provided.

An analogue of the Eneström-Kakeya theorem is established for these polynomials.

Abstract

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It can also be defined for other homogeneous polynomials not corresponding to existing codes. If the homogeneous polynomial is invariant under the MacWilliams transform, then its zeta function satisfies a functional equation and we can formulate an analogue of the Riemann hypothesis. As far as existing codes are concerned, the Riemann hypothesis is believed to be closely related to the extremal property. In this article, we show there are abundant polynomials invariant by the MacWilliams transform which satisfy the Riemann hypothesis. The proof is carried out by explicit construction of such polynomials. To prove the Riemann hypothesis for a certain class of invariant polynomials, we establish an analogue of the Enestr"om-Kakeya theorem.