On sums of $k$-th powers with almost equal primes

TL;DR

This paper establishes an asymptotic formula for the number of solutions to sums of k-th powers of primes that are almost equal, extending previous results by enlarging the range of the number of primes involved.

Contribution

It introduces a new approach that avoids exponential sums and Vinogradov mean value theorems, utilizing results from Kumchev and Liu to improve the effective range of s.

Findings

Asymptotic formula for solutions with almost equal primes

Enlarged range of s compared to previous methods

Avoidance of exponential sums and Vinogradov mean value theorems

Abstract

For "almost all" sufficiently large $N,$ satisfying necessary congruence conditions and $k\geq 2$, we show that there is an {\bf asymptotic formula} for the number of solutions of the equation \begin{align*} \begin{split} &N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k}, \\ &\left|p_{i}-( N/s)^{1/k}\right|\leq (N/s)^{\theta/k},\ (1\leq i\leq s) \end{split} \end{align*} with \begin{align*} s\geq \frac{k(k+1)}{2}+1\ \ \textup{and}\ \ \theta\geq {\bf 2/3}+\varepsilon. \end{align*} This enlarges the effective range of $s$ for which can be obtained by the method of M\"{a}tomaki and Xuancheng Shao \cite{MS}. [Discorrelation between primes in short intervals and polynomial phase, Int. Math. Res. Not. IMRN 2021, no. 16, 12330-12355.] The idea is to avoid using the exponential sums (1.2) and Vinogradov mean value theorems in Lemma 2.4 simultaneously. And the main new ingredient is from Kumchev and Liu \cite{KL} (see Lemma 2.2).