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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.

MSC:

11N13 Primes in congruence classes
11N69 Distribution of integers in special residue classes
11Y35 Analytic computations
11R44 Distribution of prime ideals
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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).

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