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A Variant of Lehmer's Conjecture, II: The CMcase


Published:20110117
Printed: Apr 2011
Sanoli Gun,
The Institute of Mathematical Sciences, CIT Campus, Taramani, India
V. Kumar Murty,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
Abstract
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 $n$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 $(n, a(n)) = 1$, where $a(n)$ is the $n$th Fourier coefficient of
a normalized Hecke eigenform $f$ of weight $2$ with rational integer
Fourier coefficients and having complex multiplication.