##
**Chebyshev’s bias.**
*(English)*
Zbl 0823.11050

Chebyshev noted that the primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. The authors study this phenomenon and its generalizations.

Let \(\pi (x; q, a_ i)\) count the primes \(p\leq x\) with \(p\equiv a\pmod q\), and consider the set of \(x\) such that \(\pi (x; q, a_ 1)> \pi(x; q, a_ 2)> \cdots >\pi (x; q, a_ r)\). Besides these sets, the case of the set of \(x\) for which \(\pi (x)\) exceeds the logarithmic integral, and the case of the excess of primes that are quadratic residues \(\pmod q\) over those that are non-residues are also considered. The logarithmic density for such a set is defined and various machineries are introduced in order to investigate these densities and biases. Assuming The Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (concerning the linear independency of the zeros of \(L\)-functions) the authors characterize exactly those moduli and residue classes for which the bias is present.

The precise statements of the theorems involve too many technical definitions to be given here. Results of numerical investigations as well as generalizations of the bias to the distribution of primes in ideal classes in number fields are also given.

Let \(\pi (x; q, a_ i)\) count the primes \(p\leq x\) with \(p\equiv a\pmod q\), and consider the set of \(x\) such that \(\pi (x; q, a_ 1)> \pi(x; q, a_ 2)> \cdots >\pi (x; q, a_ r)\). Besides these sets, the case of the set of \(x\) for which \(\pi (x)\) exceeds the logarithmic integral, and the case of the excess of primes that are quadratic residues \(\pmod q\) over those that are non-residues are also considered. The logarithmic density for such a set is defined and various machineries are introduced in order to investigate these densities and biases. Assuming The Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (concerning the linear independency of the zeros of \(L\)-functions) the authors characterize exactly those moduli and residue classes for which the bias is present.

The precise statements of the theorems involve too many technical definitions to be given here. Results of numerical investigations as well as generalizations of the bias to the distribution of primes in ideal classes in number fields are also given.

Reviewer: Peter Shiu (Loughborough)

### MSC:

11N13 | Primes in congruence classes |

11N69 | Distribution of integers in special residue classes |

11Y35 | Analytic computations |

11R44 | Distribution of prime ideals |

### Keywords:

comparative number theory; generalized Riemann hypothesis; grand simplicity hypothesis; bias; distribution of primes in ideal classes
PDFBibTeX
XMLCite

\textit{M. Rubinstein} and \textit{P. Sarnak}, Exp. Math. 3, No. 3, 173--197 (1994; Zbl 0823.11050)

### Online Encyclopedia of Integer Sequences:

Where prime race 4m-1 vs. 4m+1 is tied.Where the prime race 3k-1 vs. 3k+1 changes leader.

Indices of primes at which the prime race 4k-1 vs. 4k+1 is tied.

Values of n for which pi_{4,3}(p_n) - pi_{4,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{8,7}(p_n) - pi_{8,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{8,7}(p) - pi_{8,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{24,13}(p_n) - pi_{24,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{24,13}(p) - pi_{24,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{3,2}(p_n) - pi_{3,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{3,2}(p) - pi_{3,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{12,5}(p_n) - pi_{12,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{12,5}(p) - pi_{12,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{12,7}(p_n) - pi_{12,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{12,7}(p) - pi_{12,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{8,5}(p_n) - pi_{8,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{8,5}(p) - pi_{8,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{24,17}(p_n) - pi_{24,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{24,17}(p) - pi_{24,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Values of n for which pi_{24,19}(p_n) - pi_{24,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Primes p for which pi_{24,19}(p) - pi_{24,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

### References:

[1] | Armitage J. V., Inv. Math. 15 pp 199– (1972) · Zbl 0233.12006 · doi:10.1007/BF01404125 |

[2] | Bays C., Math. Comp. 32 pp 571– (1978) |

[3] | DOI: 10.1007/978-1-4615-7386-9 · doi:10.1007/978-1-4615-7386-9 |

[4] | Davenport H., Multiplicative Number Theory,, 2. ed. (1980) · Zbl 0453.10002 |

[5] | Davidoff G., ”Some experimental results on a generalization of Littlewood’s Theorem” (1994) |

[6] | Dirichlet L., Abh. König. Preuss. Akad. 34 pp 45– (1837) |

[7] | Heath-Brown R., Acta Arith. 60 (4) pp 389– (1992) |

[8] | Hooley C., J. London Math. Soc. (2) 16 pp 1– (1977) · Zbl 0377.10023 · doi:10.1112/jlms/s2-16.1.1 |

[9] | Ingham A. E., The Distribution of Prime Numbers (1932) · Zbl 0006.39701 |

[10] | Kaczorowski J., ”The boundary values of generalized Dirichlet series and a problem of Chebyshev” · Zbl 0790.11066 |

[11] | Knapowsky S., Acta. Math. Acad. Sci. Hungar. 13 pp 299– (1962) · Zbl 0111.04506 · doi:10.1007/BF02020796 |

[12] | Kueh K. L., J. reine angew. Math. 385 pp 1– (1988) |

[13] | Lang S., Algebraic Number Theory (1970) · Zbl 0211.38404 |

[14] | DOI: 10.1112/jlms/s1-32.1.56 · Zbl 0086.03501 · doi:10.1112/jlms/s1-32.1.56 |

[15] | Lévy P., C. R. Acad. Sci. Paris 175 pp 854– (1922) |

[16] | Littlewood J. E., C. R. Acad. Sci. Paris 158 pp 1869– (1914) |

[17] | Littlewood J. E., Proc. London Math. Soc. (2) 27 pp 358– (1928) · doi:10.1112/plms/s2-27.1.358 |

[18] | Montgomery, H. L. ”The Zeta Function and Prime Numbers”. Proceedings of the Queen’s Number Theory Conference. 1979. Edited by: Ribenboim, P. pp.14–24. Kinston, Ont.: Queen’s University. [Montgomery 1980] |

[19] | Odlyzko A., ”The 1020-th zero of the Riemann Zeta Function and 70 million of its neighbours” |

[20] | Phillips R., Duke Math. J. 55 pp 287– (1987) · Zbl 0642.53050 · doi:10.1215/S0012-7094-87-05515-3 |

[21] | te Riele H. J. J., Math. Comp. 48 pp 667– (1986) |

[22] | Rubinstein M., ”Chebyshev’s bias” (1994) · Zbl 0823.11050 |

[23] | Rudnick Z., ”Zeros of principal L-functions and random matrix theory” (1994) |

[24] | Rumely R., Math Comp. 61 pp 415– (1993) · doi:10.1090/S0025-5718-1993-1195435-0 |

[25] | DOI: 10.1007/BF01418887 · Zbl 0229.13006 · doi:10.1007/BF01418887 |

[26] | Shanks D., Math. Tables (later Math. Comp.) 13 pp 272– (1959) · Zbl 0097.03001 · doi:10.2307/2002800 |

[27] | Skewes S., J. London Math. Soc. 8 pp 277– (1933) · Zbl 0007.34003 · doi:10.1112/jlms/s1-8.4.277 |

[28] | Stein E., Introduction to Fourier Analysis on Euclidean Spaces (1971) · Zbl 0232.42007 |

[29] | Watson G. N., A Treatise on the Theory of Bessel Functions,, 2. ed. (1948) |

[30] | Wintner A., ”Asymptotic Distributions and Infinite Convolutions” (1938) |

[31] | Wintner A., Amer. J. Math. 63 pp 233– (1941) · Zbl 0025.10701 · doi:10.2307/2371519 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.