The Mobius function is strongly orthogonal to nilsequences

Möbius orthogonality for the Zeckendorf sum-of-digits function

We show that the (morphic) sequence $(-1)^{s_φ(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_φ$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N} (-1)^{s_φ(n)} μ(n) = o(N)$, that is, this sequence satisfies the Sarnak conjecture.

Möbius函数与多项式的正交性及其在$\mathbb{F}_p[x]$中素数线性方程的应用

Orthogonality of the Möbius function to polynomials with applications to Linear Equations in Primes over $\mathbb{F}_p[x]$

本文证明了Möbius函数与多项式的正交性，并以此计数$\mathbb{F}_p[x]$中仿射线性方程的素数解，类比Green和Tao的工作。

Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals

Let $μ(n)$ be the Möbius function, $e(z) = \exp(2πiz)$, $x$ real and $2\leq y \leq x$. This paper proves two sequences $(μ(n))$ and $(e(n^k α))$ are strongly orthogonal in short intervals. That is, if $k \geq 3$ being fixed and $y\geq x^{1-1/4+\varepsilon}$, then for any $A>0$, we have \[ \sum_{x< n \leq x+y} μ(n) e\left(n^k α\right) \ll y(\log y)^{-A} \] uniformly for $α\in \mathbb{R}$.

Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(Φ(g(n)Γ))_{n\in\mathbb{Z}^{D}}$ and aperiodic multiplicative functions on $\mathcal{O}_{K}$, the ring of integers of $K$. Here $D=[K\colon\mathbb{Q}]$, $X=G/Γ$ is a nilmanifold, $g\colon\mathbb{Z}^{D}\to G$ is a polynomial sequence, and $Φ\colon X\to \mathbb{C}$ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold…

Möbius orthogonality for $q$-semimultiplicative sequences

We show that all $q$-semimultiplicative sequences are asymptotically orthogonal to the Möbius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.

On (not) computing the Mobius function using bounded depth circuits

Any function F : {0,...,N-1} -> {-1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines a result of Linial, Mansour and Nisan with techniques of Katai and Harman-Katai, used in their work on finding primes with specified digits.

莫比乌斯函数与$\mathbb{F}_p[t]$上多项式相位的强正交性

Möbius function is strongly orthogonal to polynomial phases over $\mathbb{F}_p[t]$

本文研究了莫比乌斯函数与$\mathbb{F}_p[t]$上多项式相位的强正交性，通过新的近似结果和强界证明了幂节省界，并获得了Gowers均匀性范数的多项式界。

On arithmetic functions orthogonal to deterministic sequences

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the charac…

Oscillating sequences, Gowers norms and Sarnak's conjecture

It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that any oscillating sequence of order $d$ is orthogonal to any $d$-nilsequence arising from the skew product on the $d$-dimensional torus $\mathbb{T}^d$. The proof yields that any oscillating sequence of higher order is orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum. however, we provide an example of oscillating sequence of higher order with large Gowers norms. …

短区间中算术函数的更高一致性 I. 所有区间

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

本文证明 μ、Λ、d_k 在短区间中相对其逼近量具有任意阶 Gowers 小性，并用于素数线性方程与多重遍历平均。

The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/Γ$ is always equidistributed in a union of closed sub-nilmanifolds of $G/Γ$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = εg'γ$, where $ε(n)$ is "smooth", $γ(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $δ$) inside some sub…

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…

闵可夫斯基问号测度

Minkowski's Question Mark Measure

研究闵可夫斯基问号函数对应的奇异连续测度，结合对数势理论和正交多项式分析其正则性及Nevai类性质，并推导正交多项式零点和Christoffel函数的渐进行为。

Pisot 数字系统中的素数终端块

Terminal Blocks of Primes in Pisot Numeration Systems

本文在 Pisot 数字系统中证明了固定终端块词的素数定理，解决了 Zeckendorf 猜想，并证明了 Tribonacci 素数定理，同时通过环面模型获得了多项统计性质。