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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.

MSC:
 11N13 Primes in congruence classes
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References:
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