On the coefficients of an eigenform

- We establish an estimate on the coefficients of the eigenforms.
- Our result is related to the well known Lehmerʼs conjecture.
- The combinatorial sieve is used in our proof.

Lehmerʼs conjecture on Ramanujanʼs tau function τ(n) asserts that τ(n)≠0 for any n⩾1. We consider a variant of Lehmerʼs conjecture. Let f(z)=∑n=1∞ane2πinz be a normalized eigenform, where the an are rational integers for all n, and let g(n) be a polynomial with integer coefficients. We bound the number of n⩽x such that an and g(n) have no common factor.