# Chebyshev's conjecture and the prime number race

@article{Ford2002ChebyshevsCA, title={Chebyshev's conjecture and the prime number race}, author={Kevin Ford and Sergei Konyagin}, journal={arXiv: Number Theory}, year={2002}, pages={67-91} }

We survey results about prime number races, that is, results about the relative sizes of prime counting functions $\pi_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.

#### 17 Citations

Chebyshev's Bias for Products of Two Primes

- Mathematics, Computer Science
- Exp. Math.
- 2010

Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers less than or equal to x that are in a given arithmetic progression modulo q and… Expand

Fujii’s development on Chebyshev’s conjecture

- Mathematics
- International Journal of Number Theory
- 2019

Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to [Formula: see text]. His conjecture is equivalent to a version of the Generalized Riemann Hypothesis.… Expand

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

- Mathematics
- Canadian Mathematical Bulletin
- 2020

Abstract We generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical… Expand

Orderings of weakly correlated random variables, and prime number races with many contestants

- Mathematics
- 2015

We investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, n-way prime number… Expand

THE PRIME NUMBER RACE AND ZEROS OF DIRICHLET L-FUNCTIONS OFF THE CRITICAL LINE: PART III

- Mathematics
- 2012

We show, for any q > 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet L-functions lying off the critical line implies that �(x;q,a) <… Expand

Inequities in the Shanks-Renyi prime number race over function fields

- Mathematics
- 2021

Fix a prime p > 2 and a finite field Fq with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring Fq[T ] such that deg(m) is large. Fix an integer r > 2, and let… Expand

Chebyshev's bias for products of irreducible polynomials

- Mathematics
- Advances in Mathematics
- 2021

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions.… Expand

Chebyshev's bias for products of $k$ primes

- Mathematics
- 2016

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$… Expand

11T55 Arithmetic theory of polynomial rings over finite fields 11N45 Asymptotic results on counting functions for algebraic and topological structures

- 2021

For any k ≥ 1, this paper studies the number of polynomials having k irreducible factors (counted with or without multiplicities) in Fq[t] among different arithmetic progressions. We obtain… Expand

Chebyshev’s bias for analytic L-functions

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2019

Abstract We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached… Expand

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