关键词: 二次Waring-Goldbach问题, 圆法, Selberg特征值猜想 , Jacquet-Langlands对应, 自守形式, 筛法

Abstract: The purpose of this paper is to briefly survey the recent progress towards the quadratic Waring-Goldbach problem. Contents of the paper are as follows: the Waring-Goldbach problem, the circle method, Hua's theorem with five almost equal prime variables, treatment of the enlarged major arcs, the major arcs in the representation by four squares of primes, mean-value estimates for Dirichlet polynomials, the minor arcs and exceptional set in the representation by four squares of primes, estimates of exponential sums over primes, sieve methods in the representation by four squares of primes, Lagrange's theorem with almost prime variables, the Linnik-Gallagher problem, new results for Hua's theorem with five almost equal prime variables, sums of three squares of almost primes, Sarnak's conjecture and ternary quadratic forms.

Key words: Selberg's eigenvalue conjecture , Jacquet-Langlands correspondence, automorphic form, sieve methods, circle method, Quadratic Waring-Goldbach problem