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

Abstract

For "almost all" sufficiently large N, satisfying necessary congruence conditions and k ≥ 2, we show that there is an asymptotic formula for the number of solutions of the equationThis enlarges the effective range of s for which can be obtained by the method of Mätomaki and Xuancheng Shao [12]. [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 [9] (see Lemma 2.2).