On Sums of Powers of Almost Equal Primes

Bin Wei, Trevor D. Wooley

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
355 Downloads (Pure)

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 $\varepsilon>0$, 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.
Original languageEnglish
Pages (from-to)1130-1162
Number of pages33
JournalProceedings of the London Mathematical Society
Volume111
Issue number5
DOIs
Publication statusPublished - 12 Nov 2015

Keywords

  • math.NT
  • 11L07
  • 11P05
  • 11P32
  • 11P55

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