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

Open Access article
 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
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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 $n$-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.
MSC Classifications: 11F11, 11F30 show english descriptions Holomorphic modular forms of integral weight
Fourier coefficients of automorphic forms
11F11 - Holomorphic modular forms of integral weight
11F30 - Fourier coefficients of automorphic forms

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