An effective analytic formula for the number of distinct irreducible factors of a polynomial

TL;DR

This paper presents an explicit analytic formula with constants for counting the distinct irreducible factors of integer polynomials, utilizing an effective version of Mertens' theorem for number fields to identify relevant primes.

Contribution

It introduces a new explicit formula and method for determining the number of irreducible factors of polynomials over integers, with practical certification via prime lists.

Findings

Provides an explicit formula with constants for factor count

Uses an effective Mertens' theorem for number fields

Yields a finite prime list certifying the factorization

Abstract

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb{Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of $f$.