Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / c / Solution

Write $\sigma =\mathrm{Re}s$. The assumed coefficient bounds giveThe comparison p-series converges exactly whenor $\mathrm{Re}s>1+(k+l)/2$. Therefore the Rankin–Selberg convolution $L(f,g,s)$ converges absolutely in the stated half-plane.