Reference Database
(references for the Prime Pages)
The Prime Pages

Search Site

How Many?


Prime Curios!
e-mail list

Prime Lists

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 ]

Item(s) in original BibTeX format

	author={G. Martin},
	title={Asymmetries in the {S}hanks-{R}{\'e}nyi prime number race},
	abstract={It has been well-observed that an inequality of the type $\pi(x;q,a) > \pi(x;q,b)$
		is more likely to hold if $a$ is a non-square modulo $q$ and $b$ is a square
		modulo $q$ (the so-called ``Chebyshev Bias''). For instance, each of $\pi(x;8,3)$,
		$\pi(x;8,5)$, and $\pi(x;8,7)$ tends to be somewhat larger than $\pi(x;8,1)$.
		However, it has come to light that the tendencies of these three $\pi(x;8,a)$
		to dominate $\pi(x;8,1)$ have different strengths. A related phenomenon
		is that the six possible inequalities of the form $\pi(x;8,a_1) > \pi(x;8,a_2)
		> \pi(x;8,a_3)$ with {$a_1,a_2,a_3$}={3,5,7} are not all equally likely---some
		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.},
	booktitle={Number theory for the millennium, II (Urbana, IL, 2000)},
	publisher={A K Peters},
	year= 2002,
	address={Natick, MA},
	mrnumber={1 956 261},

Prime Pages' Home
Another prime page by Chris K. Caldwell