Home
Search Site
Largest
Finding
How Many?
Mersenne
Glossary
Prime Curios!
email list
FAQ
Prime Lists
Titans
Submit primes

This is the Prime Pages'
interface to our BibTeX database. Rather than being an exhaustive database,
it just lists the references we cite on these pages. Please let me know of any errors you notice.References: [ Home  Author index  Key index  Search ]
 Martin2000
 G. Martin, Asymmetries in the ShanksRényi prime number race. In "Number theory for the millennium, II (Urbana, IL, 2000)," A K Peters, 2002. Natick, MA, Preprint. MR 1 956 261
Abstract:
It has been wellobserved that an inequality of the type π(x;q,a) > π(x;q,b) is more likely to hold if a is a nonsquare modulo q and b is a square modulo q (the socalled "Chebyshev Bias"). For instance, each of π(x;8,3), π(x;8,5), and π(x;8,7) tends to be somewhat larger than π(x;8,1). However, it has come to light that the tendencies of these three π(x;8,a) to dominate π(x;8,1) have different strengths. A related phenomenon is that the six possible inequalities of the form π(x;8,a_{1}) > π(x;8,a_{2}) > π(x;8,a_{3}) with {a_{1},a_{2},a_{3}}={3,5,7} are not all equally likelysome orderings are preferred over others. In this paper we discuss these phenomena, focusing on the moduli q=8 and q=12, and we explain why the observed asymmetries (as opposed to other possible asymmetries) occur.
