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

Abstract Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ ( n ) {\tau(n)} is odd if and only if n is an odd square, we only need to consider τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} for primes p and natural numbers n ≥ 1 {n\geq 1} . This is a rather delicate question. In this direction, we show that for any ϵ > 0 {\epsilon>0} and integer n ≥ 1 {n\geq 1} , the largest prime factor of τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} , denoted by P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) {P(\tau(p^{2n}))} , satisfies P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) > ( log ⁡ p ) 1 8 ⁢ ( log ⁡ log ⁡ p ) 3 8 - ϵ P(\tau(p^{2n}))>(\log p)^{\frac{1}{8}}(\log\log p)^{\frac{3}{8}-\epsilon} 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.