# A Refined Conjecture for the Variance of Gaussian Primes Across Sectors

## TL;DR

This paper refines the conjecture on the variance of Gaussian primes in sectors, incorporating lower order terms and bifurcation points, supported by both heuristic and unconditional results.

## Contribution

It introduces a refined conjecture with a power saving error term, revealing a second bifurcation point overlooked by previous RMT models.

## Key findings

- Identification of a second bifurcation point in the variance model
- Unconditional proof for small sectors consistent with the conjecture
- Application of the L-functions Ratios Conjecture to Gaussian primes

## Abstract

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.