"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 



Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory
[Applications of Symmetric Power $L$-functions](http://www.mast.queensu.ca/~murty/Murty.pdf), 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.
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