Distribution of primes represented by polynomials and Multiple Dedekind zeta functions

TL;DR

This paper proposes conjectures on the distribution of primes represented by irreducible polynomials and their relation to multiple Dedekind zeta functions, supported by numerical evidence in quadratic fields.

Contribution

It introduces new conjectures linking prime distributions to multiple Dedekind zeta functions and provides numerical validation for quadratic fields.

Findings

Conjectures relate prime distributions to multiple Dedekind zeta functions.

Numerical tests show less than 0.1% error in quadratic fields.

Data and code are publicly available on GitHub.

Abstract

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadratic fields. The conjecture for pairs of primes and multiple Dedekind zeta function over the Gaussian integers provide error less than a tenth of a percent. We also tested conjectures that compare sets of primes in a pair of different quadratic fields. Numerically, such quotients can be expressed in terms of regulators and class numbers. Some of the data, together with the code, is available on GitHub, (see \cite{Zouberou}).