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

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