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Biases in the Shanks-Rényi prime number race. (English) Zbl 0976.11041
The Shanks-Rényi prime number race concerns systems of inequalities of the form \[ \pi(x,q,a_1) >\pi(x,q,a_2) >\cdots> \pi(x,q,a_r),\tag{*} \] where, as usual, \(\pi(x,q,a)\) denotes the number of primes up to \(x\) that are congruent to \(a\pmod q\). Assuming the Riemann Hypothesis for Dirichlet \(L\)-functions \(\pmod q\) and the linear independence over \(\mathbb{Q}\) of their imaginary parts, M. Rubinstein and P. Sarnak [Exp. Math. 3, 173-197 (1994; Zbl 0823.11050)] proved that the set of \(x\) satisfying (*) has a positive logarithmic density \(\delta_{q;a_1,\dots,a_r}\). Using the same assumptions the present authors give a general formula for these densities and using it they study the behavior of \(\delta_{q;a_1,\dots,a_r}\) under permutations of the \(a_j\). In general it turns out that the densities in question are asymmetric under such permutations. There are however situations in which they remain unchanged.

11N13 Primes in congruence classes
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