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

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Set functions and measures on spaces with additional structure Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  30 January 2020

Lucile Devin
Lucile Devin*
Affiliation:
Centre de recherches mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal, Québec, H3T 1J4Canada Email: devin@crm.umontreal.ca
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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.

Keywords

Chebyshev’s biasalmost-periodic functionregularity of measure

MSC classification

Primary: 11K70: Harmonic analysis and almost periodicity
Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.) 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 837 - 849
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

This work was partially supported by a Postdoctoral Fellowship at the University of Ottawa.

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