Andrew Granville, K. Soundararajan
Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.
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In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.
Classical mean-value results of Wirsing type in analytic number theory are established under weaker than classical conditions.
We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic $L$-function on $GL_m (m \geq 2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic $L$-function and polynomial nilsequences also has logarithmic decay.
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.
On the geometric mean of the values of positive multiplicative arithmetical functions
本文针对满足特定渐近条件的正强乘性函数，推导了其值几何平均值的渐近展开式。
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.
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.
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…
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
Using elementary means, we prove several identities involving the Möbius function, generalizing in the multidimensional case well-known formulas coming from convolution arguments.
Moyennes effectives de fonctions multiplicatives complexes
本文为一类广泛的乘法算术函数建立有效均值估计，给出Wirsing经典估计的定量版本并推广Halász的结果，同时应用于伪饰函数均值差、素因子局部分布及加性函数加权分布。
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…
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 compute the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in $[0,1]$ whose binary expansion $(x_k)$ satisfies $x_k x_{2k}=0$ for all $k\ge 1$, and show that it is smaller than the Minkowski dimension.
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
Mean value theorems in multiplicative systems and joint ergodicity of additive and multiplicative actions
本文研究乘法系统中的加法遍历平均，推广 Halász 均值定理，刻画有限生成乘法系统的伪装动力学行为，并证明加法与乘法作用在无局部障碍时联合遍历，进而得到 Szemerédi 定理特殊情形的精细化组合结果。
Mean values of multiplicative functions and applications to the distribution of the sum of divisors
本文在一般假设下建立乘法函数均值的一致幂次节省界，并据此研究 σ(n) 在模 q 互素剩余类中的等分布，推广并优化了 Śliwa 的结果。
Mean values of multiplicative functions and applications to residue-class distribution
本文给出了乘性函数部分和的一致界，并据此对 Alladi-Erdős 函数和 Euler totient 函数在不同模数下的剩余类分布给出了近乎最优的估计与均匀性量化分析。
$L^1$ means of exponential sums with multiplicative coefficients. II
本文研究带乘法系数的指数和 $L^1$ 均值，证明若其均值较小且 $L^2$ 范数较大，则函数在素数上近似于低导子乘法特征，结果关于参数一致且精确。
Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More precisely, let Gamma_N(S) = {1/N sum_{n <= N} f(n): f in F(S)} and Gamma(S) = lim_{N -> infinity} Gamma_N(s). The authors call Gamma(S) the spectrum of the set S, and study its properties.
$L^1$ means of exponential sums with multiplicative coefficients. I
本研究改进了带Liouville函数和Möbius函数系数的指数和$L^1$范数的下界，通过利用系数与Dirichlet $L$-函数零点的联系统一两种情形的证明，并将在后续论文中推广至一般乘性函数。
Asymptotic formula for the multiplicative function $\frac{d(n)}{k^{ω(n)}}$
本文针对固定整数 $k$，定义乘法函数 $D_{k,ω}(n)=\ rac{d(n)}{k^{ω(n)}}$，并利用初等方法研究其均值，给出渐近公式。
We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a completely multiplicative function $f : \mathbb{N} \to \{-1,1\},$ \begin{align*} \limsup_{x\to\infty}\Big|\sum_{n\leq x}μ^2(n)f(n)\Big|=\infty. \end{align*} This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplica…
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…
Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。
Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors.The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem 3). The same i…
We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a conjecture of Helson. We also prove asymptotics for the even integer moments. The latter have also been obtained very recently and independently by Heap and Lindqvist. Our proofs involve general lower bound techniques for random multiplicative functions, mean value results for multiplicative functions in several variables, and some calculations with Birkhoff polytopes.
We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We …
Let $1/2\leqβ<1$, $p$ be a generic prime number and $f_β$ be a random multiplicative function supported on the squarefree integers such that $(f_β(p))_{p}$ is an i.i.d. sequence of random variables with distribution $\mathbb{P}(f(p)=-1)=β=1-\mathbb{P}(f(p)=+1)$. Let $F_β$ be the Dirichlet series of $f_β$. We prove a formula involving measure-preserving transformations that relates the Riemann $ζ$ function with the Dirichlet series of $F_β$, for certain values of $β$, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted…
Multiplicative chaos measure for multiplicative functions: the $L^1$-regime
本文针对一大类乘性函数 $f$，利用改进的二阶矩方法和近似 Girsanov 定理，在整个 $L^1$-regime 下构建了由 $αf$ 的 Dirichlet 级数产生的乘性混沌测度，并证明了其与近似方案无关，进而导出了 $αf$ 归一化和的广义中心极限定理。
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and some soft number theoretic input that comes in the form of an orthogonality criterion of Kátai. We use variants of this structure theorem to derive applications of number theoretic and combinatorial flavor: $(i)$ we give simple necessary and sufficient conditions for the Gowe…
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