A random polynomial with multiplicative coefficients is almost surely irreducible

TL;DR

The paper proves that, under the assumption that the Riemann hypothesis holds for Dedekind zeta functions, a random polynomial with multiplicative coefficients drawn from dependent $\pm1$ variables is irreducible over $\mathbb{Z}$ with probability $1- O(d^{-1/2+\varepsilon})$. The authors adapt the BV19 strategy by linking irreducibility to the average number of roots in random finite fields and establishing equidistribution of the polynomial’s values $P(a)$ for most residues $a$ via conditioning on small primes and exploiting many disjoint arithmetic progressions among primes. They develop a robust equidistribution framework for sums of the form $\sum X_i a^i$ in $\mathbb{F}_q$, extend it to the pair $(P(a),P'(a))$, and deduce tight control on the expected number of roots and double roots, which then yields irreducibility with high probability and a bound on the possibility of the polynomial being a proper power. The results contribute to understanding irreducibility in models with dependent coefficients and connect number-theoretic distribution results (via RH for zeta functions and GS-type L-function assumptions) to probabilistic polynomial irreducibility phenomena, with corollaries under Legendre-symbol distributions as a further application.

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 $1-O(d^{-1/2+\varepsilon})$.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof : Proof of Proposition \ref{['pr:number-distinct-factors']}
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• proof