Non-lacunarity of Fourier coefficients

Question

Let $f=\sum_{n\geq 1}a_n q^n$ be a non-CM newform of weight $\geq 2$. It is a classical result of Serre that if $A=\{p-\text{prime}~|~a_p=0\}$ then for $x\geq 2$: $$|A\cap [1,x]|\ll \frac{x}{(\log x)^{3/2-\delta}}\text{, for any $\delta>0$}.$$ How can this result be used to show that $$\sum_{p\in A}{\frac{1}{p}}<\infty ?$$ Is the same true for Maass forms?

Answer 1

Consider the subsum over the primes $p\in A$ satisfying $2^k\leq p<2^{k+1}$. By Serre's result, this subsum is $\ll k^{-5/4}$, upon choosing $\delta:=1/4$. Summing these subsums over the positive integers $k$, the result follows.

Regarding your second question: I believe Serre's result has not been extended to Maass forms.