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

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 π(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 , . . . , 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 π(x; q, a 0 ) > π(x; q, a 1 ) > . . . > π(x; q, a D ) are simultaneously satisfied admits a logarithmic density.

Erratum: Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

“…. , yγ N ) : y ∈ R}/(2πZ) N is connected unconditionally : it is the continuous image of the connected set R. This last remark corrects [De,Rem. 2.3.…”

“…, γ N Q . In particular, the hypothesis is stronger than the one in [De,Th. 2.2], but still weaker than the full Linear Independence.…”

“…Finally, the hypothesis in Theorem 1 is now too strong to deduce [De,Cor. 2.6], we should thus consider this statement as not proved.…”

“…2.2]. This is also the cause of the dierence between the continuous case ( [De,Th. 1.2]) and the discrete case.…”

Inequities in the Shanks–Renyi prime number race over function fields

“…Cha, Fiorilli, and Jouve studied the prime number race for elliptic curves over the function field of a proper, smooth, and geometrically connected curve over a finite field. Further related subjects have been studied since then in [7], [6], [4], [16], and [1].…”