On Sums of Powers of Almost Equal Primes

Abstract

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 $\epsilon >0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{{\theta }_k+\epsilon }$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.

38 pages; in version 2 we have corrected a significant oversight in section 4 of the original version, leading to a slight adjustment of the admissible exponents for larger k

References in corpus (2)

• Rational solutions of pairs of diagonal equations, one cubic and one quadratic
• Waring-Goldbach problem in short intervals

Cited by in corpus (1)

• Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals