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A Variant of Lehmer’s Conjecture, II: The CM-case

Published online by Cambridge University Press:  20 November 2018

Sanoli Gun  and
V. Kumar Murty
Sanoli Gun
Affiliation:
The Institute of Mathematical Sciences, CIT Campus, Taramani, India email: sanoli@imsc.res.in
V. Kumar Murty
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4 email: murty@math.toronto.edu
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Abstract

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Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n\text{-th}$ Fourier coefficient of $f$. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers $n$ for which $\left( n,\,a\left( n \right) \right)\,=\,1$, where $a\left( n \right)$ is the $n\text{-th}$ Fourier coefficient of a normalized Hecke eigenform $f$ of weight 2 with rational integer Fourier coefficients and having complex multiplication.

Keywords

11F1111F30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 2 , 01 April 2011 , pp. 298 - 326
Copyright
Copyright © Canadian Mathematical Society 2012

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