Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract

We study higher uniformity properties of the Möbius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\epsilon }\le H\le X^{1-\epsilon }$ for a fixed constant $0\le \theta <1$ and any $\epsilon >0$. More precisely, letting ${\Lambda }^{♯}$ and $d_k^{♯}$ be suitable approximants of $\Lambda$ and $d_k$ and ${\mu }^{♯}=0$, we show for instance that, for any nilsequence $F(g(n)\Gamma )$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $\theta =5/8$ and $f\in \{\Lambda ,\mu ,d_k\}$ or $\theta =1/3$ and $f=d_2$. As a consequence, we show that the short interval Gowers norms $\Vert f-f^{♯}{\Vert }_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,\theta$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

103 pages; Some typo fixes and a slight fix in proof of Proposition 2.14 compared to the published version, acknowledgment added

References in corpus (1)

• A note on the Liouville function in short intervals

Cited by in corpus (2)

• Bracket words along Hardy field sequences
• Pointwise convergence of bilinear polynomial averages over the primes