On the coefficients of an eigenform (Q2637455)

In analytic number theory, the modular and cusp forms are not only very important objects, but also have many applications. Let \[ \begin{aligned} \Delta (z) &:=\sum_{n=0}^{\infty }\tau (n)q^{n} \\ &=q\prod_{j=1}^{\infty }(1-q^{j})^{24}, \end{aligned} \] where \(\tau (n)\) denotes the Ramanujan's and \(q=e^{2\pi iz}\) with \(\text{Im}(z)>0\). The well known Lehmer's conjecture asserts that \[ \tau (n)\neq 0 \] for any \(n\geq 1\). This conjecture has been open for a long time and seems quite difficult. More generally, one can consider the nonvanishing of the Fourier coefficients of a Hecke eigenform. In a recent work by\textit{V. K. Murty} [J. Number Theory 123, No. 1, 80--91 (2007; Zbl 1106.11015)], a variant of Lehmer's conjecture is considered and some bounds are obtained. Let \(f(z) = \sum_{n=1}^\infty a_n e^{2\pi i n z}\) be normalized eigenform, where the \(a_n\) are rational integers for all \(n\), and let \(g(n)\) be a polynomial with integer coefficients. The author obtains bounds on the number of \(n \leq x\), such that \(a_n\) and \(g(n)\) have no common factor.