Hua's theorem with \(s\) almost equal prime variables
Let \(k\) be an integer with \(2 \leq k \leq 10\), and let \(s=2^k+1\). When \(p\) is prime, let \(\tau\) be the exact power of \(p\) dividing \(k\), and define \(\gamma=\tau+2\) when \(p=2\) and \(\tau > 0\) and \(\gamma=\tau+1\) otherwise. Finally, write \(R=\prod_{(p-1)|k} p^{\gamma}\). It is a classical theorem of Hua that the equation \(N=p_1^k + \dots + p_s^k\) has solutions in primes \(p_j\) for all sufficiently large integers \(N \equiv s \pmod{R}\). Over the past 15 years, several authors have attempted to refine this result to include a condition of the shape \(|p_j - (N/s)^{1/k}| \leq N^{1/k-\Delta+\varepsilon} \;(j=1, \dots, s)\), thus ensuring the existence of solutions with the variables ``almost equal''. For example, \textit{X. Meng} [Pure Appl. Math. 18, No. 1, 5--12 (2002; Zbl 1010.11057)] showed that one can take \(\Delta^{-1}=2k[(k-1)2^{2k-2}+1]\) on GRH, while \textit{J. Liu, G. Liu}, and \textit{T. Zhan} [Sci. China, Ser. A 49, No. 5, 611--619 (2006; Zbl 1106.11033)] proved this unconditionally in the case \(k=2\). The authors of the paper under review establish Meng's result unconditionally for \(2 \leq k \leq 10\). The proof uses the iterative techniques of \textit{J. Liu} [Q. J. Math. 54, No. 4, 453--462 (2003; Zbl 1080.11071)], together with zero-density estimates for Dirichlet \(L\)-functions, to handle the major arcs in the Hardy-Littlewood method.
scientific article; zbMATH DE number 5601611
scientific article