On Sums of Powers of Almost Equal Primes

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$, $\theta_3=\frac{4}{5}$ and $\theta_k=\frac{5}{6}$ $(k\ge 4)$. Suppose that $n$ satisfies the necessary congruence conditions, and put $X=(n/s)^{1/k}$. We show that whenever $s>s_k$ and $\varepsilon>0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{\theta_k+\varepsilon}$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.