论文标题
算术函数的较高均匀性在短时间I.所有间隔
Higher uniformity of arithmetic functions in short intervals I. All intervals
论文作者
论文摘要
我们研究Möbius函数$μ$，von mangoldt函数$λ$的较高均匀性能，而除数函数$ d_k $在短时间内$（x，x+h] $（x，x+h] $，带有$ x^{θ+\ \ \ \ \\ varepsilon} $ \ varepsilon> 0 $。 更准确地说，让$λ^\ sharp $和$ d_k^\ sharp $是$λ$和$ d_k $和$ d_k $和$μ^\ sharp = 0 $的合适近似，我们表明，对于任何nilsequence $ f（g（n）γ）$ f（g（n）γ）\ ll h \ log^{ - a} x \] $θ= 5/8 $和$ f \ in \ {λ，μ，μ，d_k \} $或$θ= 1/3 $和$ f = d_2 $。 As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,θ$. 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短时间的素数汇聚在$ l^2 $中。 我们的创新包括将多参数NILSequence等分定理用于控制类型$ II $ sums，以及将双曲线附近的基本分解为算术进程，以控制类型的$ I_2 $ sums。
We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$. More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,θ$. 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.
