A random polynomial with multiplicative coefficients is almost surely irreducible

TL;DR

Under the assumption of the Riemann hypothesis for Dedekind zeta functions, this paper proves that a random polynomial with multiplicative ±1 coefficients is almost surely irreducible as degree increases.

Contribution

It establishes a probabilistic irreducibility result for polynomials with multiplicative ±1 coefficients under a key number-theoretic hypothesis.

Findings

Probability of reducibility decreases as degree increases

Almost sure irreducibility under the Riemann hypothesis assumption

Quantitative bounds on irreducibility probability

Abstract

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability $1-O(d^{-1/2+\varepsilon})$.