On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms.

Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ (n) is odd if and only if n is an odd square, we only need to consider τ ⁢ (p 2 ⁢ n) for primes p and natural numbers n ≥ 1 . This is a rather delicate question. In this direction, we show that for any ϵ > 0 and integer n ≥ 1 , the largest prime factor of τ ⁢ (p 2 ⁢ n) , denoted by P ⁢ (τ ⁢ (p 2 ⁢ n)) , satisfies P ⁢ (τ ⁢ (p 2 ⁢ n)) > (log ⁡ p) 1 8 ⁢ (log ⁡ log ⁡ p) 3 8 - ϵ for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. [ABSTRACT FROM AUTHOR]