Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes - Archive ouverte HAL Access content directly
Journal Articles Canadian Mathematical Bulletin Year : 2020

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

Lucile Devin

Abstract

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 case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$ . For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$ , assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

Dates and versions

hal-03345852 , version 1 (15-09-2021)

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Cite

Lucile Devin. Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes. Canadian Mathematical Bulletin, 2020, 63 (4), pp.837-849. ⟨10.4153/S0008439520000089⟩. ⟨hal-03345852⟩

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