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Prime number races. (English) Zbl 1139.11037
This nice paper gives a survey on comparative prime number theory. Starting from Chebyshev’s observation of a bias (primes of the form $$p\equiv 3 \bmod 4$$ are more frequent than those which are $$1 \bmod 4$$) and Littlewood’s result that infinitely often the Chebyshev bias is reversed, the authors take the reader step by step to the most recent results. In particular the surprising results by M. Rubinstein and P. Sarnak “Chebyshev’s bias” [Exp. Math. 3, No. 3, 173–197 (1994; Zbl 0823.11050)] are explained, which for the first time makes Chebyshev’s observation precise, namely that he is right, most of the time. This result is subject to well known but unproven conjectures. Further the extensions by A. Feuerverger and G. Martin, “Biases in the Shanks-Rényi prime number race” [Exp. Math. 9, No. 4, 535–570 (2000; Zbl 0976.11041)] to more general prime number races comparing more than two residue classes are discussed. It is also explained how these modern results actually grew out of an “Research experience for undergraduates” project. The paper can be recommended for similar projects to generate further research.

##### MSC:
 11N13 Primes in congruence classes 11N05 Distribution of primes 11Y35 Analytic computations
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