Hua's theorem on five squares of primes (Q2199996)

Let \(\Lambda(n)\) be the von Mangoldt function, and put \[ R_s(N)=\sum_{N=n_1^2+\dots+n_s^2}\Lambda(n_1)\cdots\Lambda(n_s). \] Also, let \(\mathfrak{S}_s(N)\) be the singular series associated to the representation of \(N\) into sum of \(s\) squares of primes. In the paper under review, the author shows that for large \(N\), \[ R_5(N)=\frac{\pi^2}{24}\,\mathfrak{S}_5(N)N^\frac{3}{2}+O\left(N^\frac{3}{2}\log^{-A}N\right), \] where \(A>0\) is arbitrary and the implied constant depends on \(A\) only. The proof based on an estimate of the weighted exponential sum \[ \sum_{n\leq N}\mu(n)e^{2\pi i f(n)}, \] where \(f\) is a real polynomial in \(n\). Applying the above approximation, the author gives a new proof of Hua's theorem asserting that each large \(N\equiv 5\pmod{24}\) can be written as a sum of five squares of primes.