A Variant of Lehmer's Conjecture in the CM Case

Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is the Ramanujan $\tau$-function. This is equivalent to the assertion that $\tau(n) \neq 0$ for any $n$. A related problem is to find the distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ } (\text{mod } p)$. These are open problems. However, the variant of estimating the number of integers $n$ for which $n$ and $\tau(n)$ do not have a non-trivial common factor is more amenable to study. More generally, let $f$ be a normalized eigenform for the Hecke operators of weight $k \geq 2$ and having rational integer Fourier coefficients $\{a(n)\}$. It is interesting to study the quantity $(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients $a(n)$ \begin{equation} \{ n \leq x \text { } | \text{ } (n,a(n))=1\} = \displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}} \end{equation} for some constant $U_f$. We extend this result to higher weight forms. \\ We also show that \begin{equation} \{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll \displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log \log x}}. \end{equation}Ph