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Orderings of weakly correlated random variables, and prime number races with many contestants

Abstract

We investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, n-way prime number races modulo q where the biases do not dissolve when \(n, q\rightarrow \infty \). We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of q, whereas previous methods could only allow a power of \(\log q\). The proofs use harmonic analysis related to the Hardy–Littlewood circle method to control the average size of correlations in prime number races. They also use various probabilistic tools, including an exchangeable pairs version of Stein’s method, normal comparison tools, and conditioning arguments. In the process we derive some general results about orderings of weakly correlated random variables, which may be of independent interest.

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Acknowledgements

We would like to thank Daniel Fiorilli for a helpful clarification of the discussion of one of our results, and the referee for their comments and corrections.

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Correspondence to Youness Lamzouri.

Additional information

Adam J. Harper is supported by a research fellowship at Jesus College, Cambridge; and, when this work was started, by a postdoctoral fellowship from the Centre de Recherches Mathématiques, Montréal. Youness Lamzouri is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

Appendix A: Sketch proof of Lemma 4.1

Appendix A: Sketch proof of Lemma 4.1

In this appendix we very briefly indicate how to deduce Lemma 4.1 from Theorem 2.1 of Reinert and Röllin [30].

Indeed, the exchangeable pair construction and calculations used to deduce Central Limit Theorem 1, in an appendix of the preprint [10], transfer directly to this situation and imply that

$$\begin{aligned} |\mathbb {E}h(W) - \mathbb {E}h(Z)|\ll & {} |h|_2 \sum _{a,b \in \mathcal {A}} \sqrt{\sum _{i=1}^{m} |c_{i}(a)|^{2}|c_{i}(b)|^{2} \mathbb {E}|V_{i}|^{4}}\\&+ |h|_3 \sum _{i=1}^{m} \mathbb {E}|V_{i}|^{3} \left( \sum _{a \in \mathcal {A}} |c_{i}(a)| \right) ^{3}. \end{aligned}$$

In Lemma 4.1 we assume the uniform fourth moment bound \(\mathbb {E}|V_{i}|^4 \le K^{4}/m^{2}\), and so the first term is

$$\begin{aligned} \ll \frac{|h|_2 K^{2}}{m} \sum _{a,b \in \mathcal {A}} \sqrt{\sum _{i=1}^{m} |c_{i}(a)|^{2}|c_{i}(b)|^{2}} , \end{aligned}$$

which is acceptable for Lemma 4.1. We also note that

$$\begin{aligned} \mathbb {E}|V_{i}|^{3} \le \frac{K^{3}}{m^{3/2}} + \mathbb {E}|V_{i}|^{3} \mathbf 1 _{|V_i| > K/\sqrt{m}} \le \frac{K^{3}}{m^{3/2}} + \frac{\sqrt{m}}{K} \mathbb {E}|V_{i}|^{4} \le 2\frac{K^{3}}{m^{3/2}}, \end{aligned}$$

and therefore the second term above is

$$\begin{aligned} \ll \frac{|h|_3 K^{3}}{m^{3/2}} \sum _{i=1}^{m} \left( \sum _{a \in \mathcal {A}} |c_{i}(a)| \right) ^{3} , \end{aligned}$$

as required for Lemma 4.1. \(\square \)

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Harper, A.J., Lamzouri, Y. Orderings of weakly correlated random variables, and prime number races with many contestants. Probab. Theory Relat. Fields 170, 961–1010 (2018). https://doi.org/10.1007/s00440-017-0800-2

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  • DOI: https://doi.org/10.1007/s00440-017-0800-2

Mathematics Subject Classification

  • Primary 11N13
  • Secondary 11N69
  • 11M26