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LARGE BIAS FOR INTEGERS WITH PRIME FACTORS IN ARITHMETIC PROGRESSIONS

Part of: Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  15 February 2018

Xianchang Meng
Xianchang Meng*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. email xmeng13@illinois.edu
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Abstract

We prove asymptotic formulas for the number of integers at most $x$ that can be written as the product of $k~({\geqslant}2)$ distinct primes $p_{1}\cdots p_{k}$ with each prime factor in an arithmetic progression $p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$, $(a_{j},q)=1$ $(q\geqslant 3,1\leqslant j\leqslant k)$. For any $A>0$, our result is uniform for $2\leqslant k\leqslant A\log \log x$. Moreover, we show that there are large biases toward certain arithmetic progressions $(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$ 11N13: Primes in progressions 11N69: Distribution of integers in special residue classes
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 237 - 252
Copyright
Copyright © University College London 2018 

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