Fujii's development on Chebyshev's conjecture

TL;DR

This paper advances the understanding of Chebyshev's conjecture, which relates to prime distribution biases and is connected to the Generalised Riemann Hypothesis, by using detailed zero computations of Dirichlet L-functions.

Contribution

The paper further strengthens Chebyshev's conjecture through extensive computational analysis of Dirichlet L-function zeros, building on Fujii's previous work.

Findings

Enhanced bounds on Chebyshev's conjecture based on zero computations

Deeper insights into the distribution of primes in specific residue classes

Support for the conjecture's validity through computational evidence

Abstract

Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to $3\pmod 4$. His conjecture is equivalent to a version of the Generalised Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it still further using detailed computations of zeroes of Dirichlet $L$-functions.