Jianya Liu

Sums of five almost equal prime squares II

AbstractIt is proved that every large integerN≡5 (mod 24) can be written as

Small Prime Solutions of Quadratic Equations

N = p_1^2 + ... + p_5^2

Enlarged major arcs in additive problems

with each primepj satisfying

Two Linnik-type problems for automorphic L -functions

|p_j - \sqrt {N/5} | \leqslant N^{\frac{{12}}{{25}} + E}

On Lagrange's Theorem with Prime Variables

, which gives a short interval version of a classical theorem of Hua.

The exceptional set in the four prime squares problem

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2+ it, f ⊗ g) N ,g,t,e k, while a convexity bound is only k. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.