Abstract. The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos-Templier did to obtain a more efficient treatment for the counting problem.

Let q be a large prime, and χ the quadratic character modulo q. Let φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and u j a Hecke-Maass cusp form for Γ 0 (q) ⊆ SL(2, Z) with spectral parameter t j . We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL (3), such asfor any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of Lfunctions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation; and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.

Let Λ(n) be the von Mangoldt function, x real and 2 ≤ y ≤ x. This paper improves the estimate on the exponential sum over primes in short intervals S k (x, y; α) = x

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