On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan $\mathrm{\Delta }$-function, we show that for any $\epsilon >0$, there exist infinitely many natural numbers $n$ such that $\tau (p^n)$ has at least $$2^{(1-\epsilon )\frac{\log n}{\log \log n}}$$ distinct prime factors for almost all primes $p$. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of Modular forms alluded to above.