Let $f=\sum_{n\geq 1} a_nq^n$ and $g=\sum_{n\geq 1} b_nq^n$ be two distinct cusp eigenforms of same weight $k\geq 2$ with real Fourier coefficients. We normalize the coefficients by defining: $\widetilde {a_n}:=\frac{a_n}{n^{(k-1)/2}}$ and similarly $\widetilde{b_n}:=\frac{b_n}{n^{(k-1)/2}}$. I was wondering if there is a quick way to see that $$\sum_{p-prime}\frac {\widetilde{a_p}-\widetilde{b_p}}{p^s}$$ is bounded as $s\to 1^{+}$.