Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Chebyshev’s bias for composite numbers with restricted prime divisors
HTML articles powered by AMS MathViewer

by Pieter Moree;
Math. Comp. 73 (2004), 425-449
DOI: https://doi.org/10.1090/S0025-5718-03-01536-9
Published electronically: May 21, 2003

Abstract:

Let $\pi (x;d,a)$ denote the number of primes $p\le x$ with $p\equiv a(\operatorname {mod} d)$. Chebyshev’s bias is the phenomenon for which “more often” $\pi (x;d,n)>\pi (x;d,r)$, than the other way around, where $n$ is a quadratic nonresidue mod $d$ and $r$ is a quadratic residue mod $d$. If $\pi (x;d,n)\ge \pi (x;d,r)$ for every $x$ up to some large number, then one expects that $N(x;d,n)\ge N(x;d,r)$ for every $x$. Here $N(x;d,a)$ denotes the number of integers $n\le x$ such that every prime divisor $p$ of $n$ satisfies $p\equiv a(\mathrm {mod} d)$. In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, $N(x;4,3)\ge N(x;4,1)$ for every $x$.

In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.

References
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2000): 11N37, 11Y60, 11N13
  • Retrieve articles in all journals with MSC (2000): 11N37, 11Y60, 11N13
Bibliographic Information
  • Pieter Moree
  • Affiliation: KdV Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
  • MR Author ID: 290905
  • Email: moree@science.uva.nl
  • Received by editor(s): November 7, 2001
  • Received by editor(s) in revised form: May 2, 2002
  • Published electronically: May 21, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Math. Comp. 73 (2004), 425-449
  • MSC (2000): Primary 11N37, 11Y60; Secondary 11N13
  • DOI: https://doi.org/10.1090/S0025-5718-03-01536-9
  • MathSciNet review: 2034131