Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory

Question

"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to find examples when one can use the analytic properties of the symmetric power $L$-functions to establish results in analytic number theory. I found an example which is the work of Guangshi Lu (On an open problem of Sankaranarayanan) in which he solved the problem of Sankaranarayanan by establishing bounds for the sums $$S_{j}(x):=\sum_{n\leq x} \lambda_f(n^j),$$ where $f$ is a modular form for the full modular group. Are there other examples when we can use the symmetric power $L$-functions to study problems related to algebraiec and geometric results in number theory? Thank you. Khadija

Answer 1

Applications of Symmetric Power $L$-functions, a set of lecture notes by Ram Murty on number theory applications including the Sato-Tate conjecture, the Ramanujan conjecture, the Selberg eigenvalue conjecture, and Artin's conjecture on the holomorphy of non-Abelian $L$-series.

The holomorphy of symmetric power $L$-functions combined with methods of averaging from analytic number theory imply surprising results with important consequences to classical questions in analytic number theory.