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Title: Sums of three squares and automorphic $L$-functions
Abstract: Is it possible to write every integer $n$ as a sum of three squares of primes, at least if we impose some natural congruence conditions on $n$? An answer to this question depends on the behaviour of certain $L$-functions in the critical strip. I will show techniques to prove good bounds for such $L$-functions, and give various arithmetic applications.