arXiv subjects

Alexander P. Mangerel

Publications and source records attributed to Alexander P. Mangerel.

32 records · Page 2Linked to original sources

arXiv2021

Divisor-bounded multiplicative functions in short intervals

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 normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length $h\log X$, if $h = h(X) \rightarrow \infty$. We also generalize this result to sequences $\{|λ_π(n)|^2\}_n$, where $λ_π(n)$ is the $n$th coefficient of the standard $L$-function of an automorphic representation $π$ with unitary central character for $GL_m$, $m \geq 2$, provided $π$ satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments $\{|λ_f(n)|^α\}_n$ over intervals of length $h(\log X)^{c_α}$, with $c_α > 0$ explicit, for any $α> 0$, as $h = h(X) \rightarrow \infty$. Finally, we show that the (non-multiplicative) Hooley $Δ$-function has average value $\gg \log\log X$ in typical short intervals of length $(\log X)^{1/2+η}$, where $η>0$ is fixed.

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arXiv2021

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Let $ε> 0$ be sufficiently small and let $0 < η< 1/522$. We show that if $X$ is large enough in terms of $ε$ then for any squarefree integer $q \leq X^{196/261-ε}$ that is $X^η$-smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod{q}$, with $(a,q) = 1$. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096...$ was replaced by $25/36 = 0.69\bar{4}$. This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$-barrier for a density 1 set of $X^η$-smooth moduli $q$ (without the squarefree condition). Our proof appeals to the $q$-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using $p$-adic methods.

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arXiv2021

Additive functions in short intervals, gaps and a conjecture of Erdős

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of its long average. As part of this treatment, we use a variant of the Matomäki-Radziwill theorem for divisor-bounded multiplicative functions recently proven by the author. We consider two sets of applications of these methods. Our first application shows that for an additive function $g: \mathbb{N} \rightarrow \mathbb{C}$ any non-trivial savings in the size of the average gap $|g(n)-g(n-1)|$ implies that $g$ must have a small first moment, i.e., the discrepancy of $g(n)$ from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing $\log n$ as the only "almost everywhere" increasing additive function (up to constant factors). We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well-approximated by a constant times a logarithm. We also show that if $g$ is a completely additive function such that the density of the set of exceptions $\frac{1}{X}|\{n \leq X : g(n) < g(n-1)\}|$ decays like $O((\log X)^{-2-ε})$ and such that $g$ is not extremely large too often on the primes (in a precise sense), then $g$ is identically equal to a constant times a logarithm.

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arXiv2021

Multiplicative functions that are close to their mean

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 multiplicative function $f : \mathbb{N} \to \mathbb{C}$ satisfies \begin{align*} \sum_{n\leq x}f(n)=cx+O(1) \end{align*} with $c\neq 0$ if and only if $f(p)=1$ for all but finitely many primes and $|f(p)|<1$ for the remaining primes. This answers a question of Ruzsa. For the case $c = 0,$ we show, under the additional hypothesis $$\sum_{p }\frac{1-|f(p)|}{p} < \infty,$$ that $f$ has bounded partial sums if and only if $f(p) = χ(p)p^{it}$ for some non-principal Dirichlet character $χ$ modulo $q$ and $t \in \mathbb{R}$ except on a finite set of primes that contains the primes dividing $q$, wherein $|f(p)| < 1.$ This provides progress on another problem of Ruzsa and gives a new and simpler proof of a stronger form of Chudakov's conjecture. Along the way we obtain quantitative bounds for the discrepancy of the generalized characters improving on the previous work of Borwein, Choi and Coons.

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arXiv2019

Short Character Sums and the Pólya-Vinogradov Inequality

We show in a quantitative way that any odd character $χ$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the Pólya-Vinogradov inequality for $χ$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t} χ(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q)$$ then for any $ε> 0$ one may exhibit cancellation in partial sums of $χ$ on the interval $[1,t]$ whenever $t > q^ε$, i.e.,$$\sum_{n \leq t} χ(n) = o_{q \rightarrow \infty}(t) \text{ for all $t > q^ε$.}$$ This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing $g$ exhibit cancellation in short sums then the Pólya-Vinogradov inequality can be improved for all odd primitive characters of order $g$. Some applications are also discussed.

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arXiv2018

On the Orbits of Multiplicative Pairs

We characterize all pairs of completely multiplicative functions $f,g:\mathbb{N}\to\mathbb{T}$ such that the orbit closure \[\overline{\{(f(n),g(n+1))\}_{n\ge 1}} \neq \mathbb{T}\times \mathbb{T}.\] In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.

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arXiv2018

Rigidity Theorems for Multiplicative Functions

We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function $f:\mathbb{N}\to\mathbb{T}$ we have $$\liminf_{n\to\infty}|f(n+1)-f(n)|=0.$$ Second, we settle an old conjecture due to N.G. Chudakov [Actes du ICM ({N}ice, 1970), {T}. 1, p. 487] that states that any completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ that: a) takes only finitely many values, b) vanishes at only finitely many primes, and c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erdős Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character $χ$ mod $q$ then $f(n) = χ'(n)n^{it}$ for all $n$, where $χ'$ is a Dirichlet character modulo $q$ and $t \in \mathbb{R}$. This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for \emph{pretentious} multiplicative functions developed earlier by the first author and Szemeredi's theorem for long arithmetic progressions.

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arXiv2017

Effective Asymptotic Formulae for Multilinear Averages of Multiplicative Functions

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 terms. \\ As an application of our formulae, we establish a \emph{local-to-global} principle for Gowers norms of multiplicative functions. We also compute the asymptotic densities of the sets of integers $n$ such that a given multiplicative function $f: \mathbb{N} \rightarrow \{-1, 1\}$ yields a fixed sign pattern of length 3 or 4 on almost all 3- and 4-term arithmetic progressions, respectively, with first term $n$.

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arXiv2017

On the Bivariate Erdős-Kac Theorem and Correlations of the Möbius Function

Let $2 \leq y \leq x$ such that $β:= \frac{\log x}{\log y} \rightarrow \infty$. Let $ω_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $μ_y(n) := μ^2(n)(-1)^{ω_y(n)}$, where $μ$ is the Möbius function. We prove that if $β$ is not too large (in terms of $x$) then for each fixed $a \in \mathbb{N}$, \begin{equation*} \sum_{n \leq x} μ_y(n)μ_y(n+a) \ll x\left(\frac{1}{\log_2 y} + e^{-\frac{1}{21}β\log β}\right). \end{equation*} This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a \emph{quantitative} bivariate analogue of the Erdős-Kac theorem regarding the distribution of the pairs $(ω(n),ω(n+a))$, where $n$ and $n+a$ both belong to any subset of the positive integers with suitable sieving properties; moreover, we show that the set of squarefree integers is an example of such a set. We end with a further application of this probabilistic result related to a problem of Erdős and Mirsky on the number of integers $n \leq x$ such that $τ(n) = τ(n+1)$.

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arXiv2017

Large odd order character sums and improvements of the Pólya-Vinogradov inequality

For a primitive Dirichlet character $χ$ modulo $q$, we define $M(χ)=\max_{t } |\sum_{n \leq t} χ(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character $χ$ we have $$M(χ)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-δ_g}(\log\log q)^{-1/4+\varepsilon},$$ where $δ_g := 1-\frac{g}π\sin(π/g)$. This improves upon the works of Granville and Soundararajan and of Goldmakher. Furthermore, assuming the Generalized Riemann hypothesis (GRH) we prove that $$ M(χ) \ll \sqrt{q} \left(\log_2 q\right)^{1-δ_g} \left(\log_3 q\right)^{-\frac{1}{4}}\left(\log_4 q\right)^{O(1)}, $$ where $\log_j$ is the $j$-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of $\log_4 q$). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.

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arXiv2016

On the Number of Restricted Prime Factors of an Integer II

Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector $\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for the quantity $|\{m \leq x: ω_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in a wide range of the parameters $k_j$ that complements the results of a previous paper of the author. This is accomplished using an extension and generalization of a theorem of Wirsing due to the author that gives explicit estimates for the ratio $\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N} \rightarrow (0,\infty)$ and $g: \mathbb{N} \rightarrow \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)| \leq f(n)$ for every $n \in \mathbb{N}$. This also allows us to conclude the validity of a probabilistic heuristic regarding $π(x;\mathbf{E},\mathbf{k})$ in the case that $k_j = (1+o(1))E_j(x)$, for each $0 \leq j \leq n$.

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arXiv2016

A Strengthening of Theorems of Halász and Wirsing

Given an arithmetic function $g(n)$ write $M_g(x) := \sum_{n \leq x} g(n)$. We extend and strengthen the results of a fundamental paper of Halász in several ways by proving upper bounds for the ratio of $\frac{|M_g(x)|}{M_{|g|}(x)}$, for any strongly multiplicative, complex-valued function $g(n)$ under certain assumptions on the sequence $\{g(p)\}_p$. We further prove an asymptotic formula for this ratio in the case that $|\text{arg}(g(p))|$ is sufficiently small uniformly in $p$. In so doing, we recover a new proof of an explicit lower mean value estimate for $M_{f}(x)$ for any non-negative, multiplicative function satisfying $c_1 \leq |f(p)| \leq c_2$ for $c_2 \geq c_1 > 0$, by relating it to $\frac{x}{\log x}\prod_{p \leq x} \left(1+\frac{f(p)}{p}\right)$. As an application, we generalize our main theorem in such a way as to give explicit estimates for the ratio $\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N} \rightarrow (0,\infty)$ and $g: \mathbb{N} \rightarrow \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)| \leq f(n)$ for every $n \in \mathbb{N}$. This generalizes a theorem of Wirsing and extends recent work due to Elliott.

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arXiv2015

On the Distribution of Integers with Restricted Prime Factors I

Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $π(x;\mathbf{E},\mathbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in $E_j$ for each $j$. Basic probabilistic heuristics suggest that $x^{-1}π(x;\mathbf{E},\mathbf{k})$, modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_0(x),\ldots,E_n(x))$, as $x \rightarrow \infty$. We prove an asymptotic formula for $π(x;\mathbf{E},\mathbf{k})$ which contradicts these heuristics in the case that for each $j$, $E_j(x)^2 \leq k_j \leq \log^{\frac{2}{3}-ε} x$ for each $j$ under mild hypotheses. As a particular application, we prove an asymptotic formula regarding integers with prime factors from specific arithmetic progressions, which generalizes a result due to Delange.

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arXiv2015

Generalizations on a Problem of Saffari

We provide a generalization of a problem first considered by Saffari and fully solved by Saffari, Erdős and Vaughan on direct factor pairs, to arbitrary finite families of direct factors, and solve it using a method of Daboussi. We end with a few related open problems.

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