Work

An extended Vinogradov’s mean value theorem

Year: 2025

Type: article

Abstract: In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov’s mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument. Let d ≥ 2 d\geq 2 be a natural number and α = ( α d , … , α 1 ) ∈ R d \boldsymbol {\alpha }=(\alpha _d,\ldots , \alpha _1)\in \mathbb {R}^d . Define the exponential sum f d ( α ; N ) ≔ ∑ 1 ≤ n ≤ N e ( α d n d + ⋯ + α 1 n ) . \begin{equation*} f_d(\boldsymbol {\alpha };N)≔\sum _{1 \leq n \leq N}e(\alpha _d n^d + \cdots + \alpha _1 n). \end{equation*} For p > 0 p>0 , consider mean values of the exponential sums I p , d ( u ; N ) ≔ ∫ [ 0 , 1 )

Source: Transactions of the American Mathematical Society

DOI: 10.1090/tran/9570

Authors: Changkeun Oh, Kiseok Yeon

Institutions: Seoul National University, University of California, Davis

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green