Dimitris Koukoulopoulos
Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n<x} f(n) \ll x/(\log x)^A$, $x>2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $μ(n)n^{it}$ for some $t$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic subgroups of the group ${\mathbb Z}_{m_1}\times\dots\times {\mathbb Z}_{m_n}$ and multivariable averages associated with the LCM function.
For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l_g(p) as p <= x ranges over primes.
Given a finitely generated multiplicative subgroup of rational numbers $Γ$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group modulo $p$. The problem was considered in the case of rank $1$ by Pomerance and Kurlberg. In the case when $Γ$ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.
A celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalizes a well known convergence result of Wirsing. The key ingredients are a recent structural result for bounded multiplicative functions proved by the authors and…
Averages of multiplicative functions along equidistributed sequences
研究非负函数沿等分布序列的平均值，建立了匹配的上下界，推广了相关平均结果。
Let $μ$ be the Möbius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $μ$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-σ}$ for any $σ> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $μ$, which corresponds to the special case $k=1$.
In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-δ}$, for some $δ>0$, and if the Dirichlet series of $f$, say $F(s)$, is such that $F(1)=0$, then we obtain that for any $ε>0$, $\sum_{p\leq x}(1+f(p))\log p\ll x^{1-δ+ε}$. Moreover, a necessary condition for the existence of such $f$ is that the Riemann zeta function $ζ(s)$ has no zeros in the half plane $Re(s)>1-δ$.
In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.
Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Halász' mean value theorem, we compute multidimensional averages of the shape $$x^{-l} \sum_{\mathbf{n} \in [x]^l} \prod_{1 \leq j \leq k} f_j(L_j(\mathbf{n}))$$ as $x \rightarrow \infty$, where $[x] := [1,x]$ and $L_1,\ldots, L_k$ are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with \emph{explicit} main and error te…
Let $λ$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_α \left | \sum_{x < n \leq x + H} λ(n) e(-αn) \right | dx = o ( X H) $$ for all $H \geq X^θ$ with $θ> 0$ fixed but arbitrarily small. Previously, this was only known for $θ> 5/8$. For smaller values of $θ$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $λ(n) Λ(n + h) Λ(n + …
Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on average. When we average over $a\leq N$, we observe a statistic of $C\mathrm{Li}(x)$. P. J. Stephens ~\cite[Theorem 1]{S} proved this statistic for $N>\exp(c_1\sqrt{\log x})$ for some positive constant $c_1$. Upon this result, we give an explicit value of $c_1$. In fact, ~\cite[Theorem 1, 3]{S} hold with $N>\exp(3.42\sqrt{\log x})$, and ~\cite[Theorem 2, 4]{S} hold with $N>\exp(4.8365\sqrt{\log x})$. Also, we improve the range of $y$, from $y…
We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $α$ on average. In the literature, the average is usually taken to be $α$ with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of $f$, developing new techniques to do so.
It is shown that the cubic nonconventional ergodic averages of any order with a bounded aperiodic multiplicative function or von Mangoldt weights converge almost surely.
Friable averages of oscillating multiplicative functions
本文研究了狄利克雷级数接近黎曼ζ函数负幂的算术函数的脆弱平均值，获得了类似Selberg-Delange方法的渐近展开，并应用于截断求和。
We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang.
Improved bounds for multiplicative functions in almost all short intervals
研究改进乘法函数短平均值的Matomäki-Radziwiłł方法，对特定函数证明最优衰减界，关键是精确处理筛法误差，还给出其他函数的改进界及局限性，应用于平均Chowla猜想获改进界。
We generalize Gel'fond's transcendence criterion to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the multiplicative group C* of the field of complex numbers.
On averages of completely multiplicative functions over co-prime integer pairs
本文研究了完全乘法函数在互质整数对上的平均值行为，推导了高斯整数上的Wirsing定理，并在多线性平均方面取得新进展。
We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences:\ First, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{R…
In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple exponential sum.
We study logarithmically averaged binary correlations of bounded multiplicative functions $g_1$ and $g_2$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_1$ or $g_2$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_j$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing…
We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log ω_m} \sum_{x_m/ω_m \leq n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n} $$ where $1 \leq ω_m \leq x_m$ are numbers going to infinity as $m \to \infty$, and $\widetilde{\lim}$ is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences $f$ are the uni…
The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,\dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely ma…
The variance of a general class of multiplicative functions in short intervals
通过傅里叶分析和特定二元形式有理点计数，建立短平均与长平均的联系，给出短区间内一般乘法函数类的渐近方差估计，并推翻van Overbeeke关于φ(n)/n方差的猜想。
On multiplicative functions with small partial sums
该研究聚焦解析数论中积性函数素数取值与平均值的反向关联，证明了除数有界且部分和小的积性函数，其素数平均取值可由对应Dirichlet级数在ℜ(s)=1上的零点虚部表示，推广了已有相关结果。
We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $φ(n)^r$, $σ(n)^r$ ($r>-1$ real), where $φ$ is Euler's totient function and $σ$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, sta…
We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3
本文研究非负乘性函数在光滑整数上的对数平均，在单侧和双侧假设下得到改进的精确估计，并据此给出了 Johnson--Selberg 素数幂筛法的有效形式。
Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limita…
We study the asymptotic behaviour of higher order correlations $$ \mathbb{E}_{n \leq X/d} g_1(n+ah_1) \cdots g_k(n+ah_k)$$ as a function of the parameters $a$ and $d$, where $g_1,\dots,g_k$ are bounded multiplicative functions, $h_1,\dots,h_k$ are integer shifts, and $X$ is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all $X$ if $g_1 \cdots g_k$ does not (weakly) pretend to be a twisted Dirichlet character $n \mapsto χ(n)n^{it}$, and behave asymptotically like a multiple of $d^{-it} χ(a)$ otherwise. This extends our earlie…
We show that the cubic nonconventional ergodic averages of any order with a bounded multiplicative function weight converge almost surely to zero provided that the multiplicative function satisfies a strong Daboussi-Delange condition. We further obtain that the Cesàro mean of the self-correlations and some moving average of the self-correlations of such multiplicative functions converge to zero. Our proof gives, for any $N \geq 2$, $$\frac1{N}\sum_{m=1}^{N}\Big|\frac1{N}\sum_{n=1}^{N} \bnu(n) \bnu(n+m)\Big| \leq \frac{C}{\log(N)^ε},$$ and $$\frac1{N^2}\sum_{n,p=1}^{N}\Big|\frac1{N}\sum_{m=1}^{…
Short Exponential Sums and Ternary Correlations of Multiplicative Functions
研究一般k-除数有界的乘法函数的三元相关性平均行为，结合圆法与加权短指数和界，利用L-函数积分矩界获得关键输入。
We extend the Matomäki-Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function $f$ in typical intervals of length $h(\log X)^c$, with $h = h(X) \rightarrow \infty$ and where $c = c_f \geq 0$ is determined by the distribution of $\{|f(p)|\}_p$ in an explicit way. We give three applications. First, we show that the classical Rankin-Selberg-type asymptotic formula for partial sums of $|λ_f(n)|^2$, where $\{λ_f(n)\}_n$ is the sequence of no…
Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation
本文为$\mathbb{R}^d$-作用证明了涉及膨胀子流形平均的新遍历定理，给出有效混合假设下的误差项定量估计，并据此研究了实系数矩阵的乘性Dirichlet可改进性，证明了几乎所有矩阵的一致逼近性，提出了Littlewood猜想的强化问题。
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski (2017).
Truncated pretentious distances of multiplicative arithmetic functions
本文研究非自命乘法函数的自相关平均收敛性，揭示其局部自命行为与素数上偏差的联系，并证明一类函数满足原始形式的 Elliott 猜想。
Let $λ_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the counting function of those integers $n$ for which $λ_1(n)\ne2$. We also give an asymptotic formula, for any even $q\ge4$, for the counting function of those integers $n$ for which $λ_1(n)=q$. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we …
In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of magnitude of $f$ and $g$. We also provide asymptotic formula for sums of the type $\sum\limits_{n \leq x} μ^2(n) G(n-h)$, where $G=g*1$ and $g$ is small. For small order of magnitudes of $f$ and $g$, we improve the error terms and make them independent of $h$.
We generalize the absolute logarithmic Weil height from elements of the multiplicative group of algebraic numbers modulo torsion, to finitely generated subgoups. The height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.
In this paper we investigate how small the density of a multiplicative basis of order $h$ can be in $\{1,2,\dots,n\}$ and in $\mathbb{Z}^+$. Furthermore, a related problem of Erd\H os is also studied: How dense can a set of integers be, if none of them divides the product of $h$ others?
We establish an interesting upper bound for the moments of truncated Dirichlet convolution of Möbius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we establish an estimate for the multiplicative energy of the set of divisors of an integer $n$.
We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $μ(n)$ in almost all intervals of the form $[x, x + ψ(x)]$ with $ψ(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^ε$-smooth numbers in intervals of the form $[x, x …
Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerful numbers. We prove that its real logarithmic spectrum takes values from $- \sqrt{2} / (4 + \sqrt{2}) = -0.26160...$ to $1$ while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from $0$ to $1$. In the course of this study, we correct a proof of Granville and Soundararajan concerning contribution of small primes in the study of mean val…
On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights
本文推广了Bourgain双重回归定理和遍历Bourgain-Sarnak定理，证明非周期$1$-有界乘性权下齐次遍历双线性平均几乎处处收敛于零，并给出关键证明要素。
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.
We prove that for an arbitrary $\varepsilon>0$ and any multiplicative subgroup $Γ\subseteq \mathbf{F}_p$, $1\ll |Γ| \le p^{2/3 -\varepsilon}$ there are no sets $B$, $C \subseteq \mathbf{F}_p$ with $|B|, |C|>1$ such that $Γ=B+C$. Also, we obtain that for $1\ll |Γ| \le p^{6/7-\varepsilon}$ and any $ξ\neq 0$ there is no a set $B$ such that $ξΓ+1=B/B$.
If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an ex…
For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals o…
In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions multiplicatively to a function on $k$-regular partitions, we show that it takes its maximum at an explicitly described small set of partitions, and can thus easily be computed. The basis for this is an extension of a classical result of Lehmer, from which an inequality for the generating function for $k$-regular partitions is deduced which seems not to have bee…
Sign changes of the partial sums of a random multiplicative function III: Average
本文研究 Rademacher 随机乘性函数部分和符号变化的平均次数，利用 Harper 的小矩估计证明了其下界至少为 (log x)(log log x)^{-1/2-ε}，并将 Erdős–Hunt 关于独立同分布情形符号变化的经典结果推广到更广泛的相依随机变量系统。
Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.
Let $λ$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \sum_{n \leq x} λ(a_1 n + b_1) λ(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ with $a_1b_2-a_2b_1 \neq 0$. In this paper we establish the logarithmically averaged version $$ \sum_{x/ω(x) < n \leq x} \frac{λ(a_1 n + b_1) λ(a_2 n+b_2)}{n} = o(\log ω(x)) $$ of the Chowla conjecture as $x \to \infty$, where $1 \leq ω(x) \leq x$ is an arbitrary function of $x$ that goes to infinity as $x \to \infty$, thus breaking the "parity barrier" for this p…
Let $λ$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} λ(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^θ \leq H \leq X$ with $0 < θ< 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences …
On Equal Consecutive Values of Multiplicative Functions
本研究在积性函数满足特定素数和发散的条件下，证明其相等非零连续值的集合对数密度为0并给出逆命题，结合Tao定理与加性组合逆理论，还应用于全纯尖点形式的傅里叶系数。
Furstenberg systems of pretentious and MRT multiplicative functions
本文研究了有界乘性函数相关的Furstenberg系统的结构性质，证明pretentious乘性函数的系统具有有理离散谱和零熵，并揭示了MRT乘性函数的新遍历现象及不同平均方式下强平稳性的差异。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。