On sums of k-th powers with almost equal primes

2022/04/16 by Wei Zhang, Zhang, Wei
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.07715

openalex publication_date 2022/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For "almost all" sufficiently large N, satisfying necessary congruence conditions and k≥ 2, we show that there is an \bf asymptotic formula for the number of solutions of the equation \beginsplit amp;N=p₁^k+p₂^k+⋯+p_s^k,
amp;|p_i-( N/s)^1/k|≤ (N/s)^θ/k, (1≤ i≤ s) \endsplit with s≥ (k(k+1))/(2)+1 \textupand θ≥ \bf 2/3+ε. This enlarges the effective range of s for which can be obtained by the method of Mätomaki and Xuancheng Shao \citeMS. [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 \citeKL (see Lemma 2.2).

Related

• Partitio Numerorum: sums of a prime and a number of k-th powers
• On a sum involving the prime counting function π(x)
• Discorrelation between primes in short intervals and polynomial phases
• On the sum of a prime power and a power in short intervals
• The sum of a prime power and an almost prime