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Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 30 January 2020, pp. 837-849
- Print publication:
- December 2020
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- Article
- Export citation
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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.
