Higher uniformity of arithmetic functions in short intervals II. Almost all intervals
成果类型：
Article
署名作者：
Matomaki, Kaisa; Radziwill, Maksym; Shao, Xuancheng; Tao, Terence; Teravainen, Joni
署名单位：
University of Turku; Northwestern University; University of Kentucky; University of California System; University of California Los Angeles; University of Cambridge
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910; 1432-1297
DOI：
10.1007/s00222-026-01408-6
发表日期：
2026-06
页码：
967-1091
关键词：
multiplicative functions
DIVISOR PROBLEM
moments
THEOREM
primes
摘要：
We study higher uniformity properties of the von Mangoldt function Lambda, the M & ouml;bius function mu, and the divisor functions d(k) on short intervals(x, x+H]for almost all x is an element of[X,2X].Let Lambda(& sharp;)and d(k)(& sharp;)be suitable approximants of Lambda and d(k) ,G/ Gamma a filtered nilmanifold, and F:G/ Gamma ->& Copf;a Lipschitz function. Then our results imply for instance that when X1/3+epsilon <= H <= Xwe have, for almost all x is an element of[X,2X], sup(g is an element of Poly(& Zopf;-> G))|& sum;(x<= x+H)(Lambda(n)-Lambda(& sharp;)(n))F(g(n)Gamma)|<< Hlog(-A)X for any fixed A>0, and that when X-epsilon <= H <= Xwe have, for almost all x is an element of[X,2X], sup(g is an element of Poly(& Zopf;-> G))|) & sum;(x<= x+H)(d(k)(n)-d & sharp;k(n))F(g(n)Gamma)|=o(Hlog(k-1)X). As a consequence, we show that the short interval Gowers norms parallel to Lambda-Lambda(& sharp;)parallel to U-s(X,X+H]and parallel to d(k) -d & sharp;k parallel to(s)(U)(X,X+H]are also asymptotically small for any fixeds in the same ranges of H. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type II estimates obtained by developing a contagion lemma for nilsequences and then using this to scale up an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.
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