Comparative prime number theory: A survey

Abstract

Comparative prime number theory is the study of the discrepancies of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime number theory, their generalizations, and an extensive list of references on both historical and current progresses. Definition 1.1. π(x; k, l) := p≤x p≡l (mod k)

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

“…[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).…”

A bias in Mertens’ product formula

“…In general, if a is a non-square modulo q and b is a square modulo q then π(x; q, a) has a strong tendency to be larger than π(x; q, b), a phenomenon which has become known as "Chebyshev's bias". For more on the history of this subject as well as recent developments, the reader is invited to consult the expository papers of Granville and Martin [6] and Martin and Scarfy [9]. Remark 1.4.…”

Extreme biases in prime number races with many contestants

“…, a n ) ∈ A n (q), we will have the ordering (1.1) π(x; q, a 1 ) > π(x; q, a 2 ) > • • • > π(x; q, a n ) for infinitely many integers x? There is now an extensive body of work investigating different aspects of this question, and the reader may consult the expository papers of Granville and Martin [6], Ford and Konyagin [5], and Martin and Scarfy [10] for fuller discussions.…”