Comparative prime-number theory. III

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

“…This observation led to the birth of comparative prime number theory, which investigates the discrepancies in the distribution of prime numbers. A central problem is the so-called "Shanks-Rényi prime number race" which is described by Knapowski and Turán [13]: let q ≥ 3 and 2 ≤ n ≤ ϕ(q) be positive integers, (where the Euler function ϕ(q) denotes the number of residue classes mod q that are coprime to q), and denote by A n (q) the set of ordered n-tuples (a 1 , a 2 , . .…”

“…, a n ) ∈ A n (q) consider a game with n players called "a 1 " through to "a n ", where at time x, the player a j has a score of π(x; q, a j ) (where π(x; q, a) denotes the number of primes p ≤ x with p ≡ a mod q). Among the questions that Knapowski and Turán asked in [13] are the following: Q1. Will each player take the lead for infinitely many integers x?…”

“…YL is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. [13], Kaczorowski [10,11,12], Feuerverger and Martin [2], Ford and Konyagin [5], Ford, Konyagin and Lamzouri [7], Fiorilli and Martin [4], Fiorilli [3], and Lamzouri [14,15]. For a complete history as well as recent developments, see the expository papers of Granville and Martin [8], Ford and Konyagin [6], and Martin and Scarfy [18] (which includes a very comprehensive list of references).…”

The Shanks–Rényi prime number race with many contestants

“…A classical problem in analytic number theory is the so-called "Shanks-Rényi prime number race" which concerns the distribution of prime numbers in arithmetic progressions. As colorfully described by Knapowski and Turán in [11], let q ≥ 3 and 2 ≤ r ≤ φ(q) be positive integers, and denote by A r (q) the set of ordered r-tuples of distinct residue classes (a 1 , a 2 , . .…”

“…An old result of Littlewood [14] shows that this is indeed true in the special cases (q, a 1 , a 2 ) = (4, 1, 3) and (q, a 1 , a 2 ) = (3, 1, 2). Since then, this problem has been extensively studied by many authors, including Knapowski and Turán [11], Bays and Hudson [1] and [2], Kaczorowski [8], [9] and [10], Feuerverger and Martin [4], Martin [15], Ford and Konyagin [6] and [7], Fiorilli and Martin [5], and the author [12] and [13].…”