arXiv subjects
Publications and source records attributed to Alexander P. Mangerel.
We construct a $1$-bounded completely multiplicative function $f$ whose logarithmically-averaged partial sums satisfy $$ \limsup_{x \rightarrow \infty} \frac{\left|\sum_{n \leq x} \frac{f(n)}{n}\right|}{1+\exp\left(\sum_{p \leq x} \frac{\text{Re}(f(p))}{p}\right)} = \infty. $$ This disproves a conjecture of Goldmakher from 2009.
We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a first application, we show that if $f$ is a completely multiplicative function taking values in $[-1,1]$ then there is a constant $c > 0$ such that for every $x \geq 3$, $$ L_f(x) := \sum_{n \leq x} \frac{f(n)}{n} > -\frac{c}{(\log x)^{1-2/π}}, $$ thus significantly improving on a 20-year-old result of Granville and Soundararajan. We also show that the exponent of $\log x$ in this result can be improved to $-1+o(1)$, as long as $f$ does not ``behave like'' the Liouville function $λ$ in a precise sense. As a second application, we show that for a Rademacher random completely multiplicative function $\mathbf{f}$, the probability that $L_{\mathbf{f}}(x)$ is negative is $O(\exp(-x^c))$ for some $c \in (0,1)$, thus establishing a previously conjectured bound. Finally, we obtain a converse theorem for small absolute values $|L_f(x)|$, and construct examples $f$ that show that it is (essentially) best possible.
Let $g \geq 3$ be fixed and odd, and for large $q$ let $χ$ be a primitive Dirichlet character modulo $q$ of order $g$. Conditionally on GRH we improve the existing upper bounds in the Pólya-Vinogradov inequality for $χ$, showing that $$ M(χ) := \max_{t \geq 1} \left|\sum_{n \leq t} χ(n) \right| \ll \sqrt{q} \frac{(\log\log q)^{1-δ_g} (\log\log\log\log\log q)^{δ_g}}{(\log\log\log q)^{1/4}}, $$ where $δ_g := 1-\tfrac{g}π\sin(π/g)$. Furthermore, we show unconditionally that there is an infinite sequence of order $g$ primitive characters $χ_j$ modulo $q_j$ for which $$ M(χ_j) \gg \sqrt{q_j} \frac{(\log\log q_j)^{1-δ_g} (\log\log\log\log\log q_j)^{δ_g}}{(\log\log\log q_j)^{1/4}}, $$ so that our GRH bound is sharp up to the implicit constant. This improves on previous work of Granville and Soundararajan, of Goldmakher, and of Lamzouri and the author.
Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss sum-like'' approximate dilation symmetry property $$ \frac{1}{N}\sum_{a \pmod{N}} |S_f(ap) - f(p)S_f(a)|^2 = o(N), $$ uniformly over all primes $p \leq N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1 \leq n < N$. As a consequence, taking $f$ to be the Liouville function we connect this exponential sums property to the location of real zeros of $L(s,χ)$ close to $s = 1$, for $χ$ the Legendre symbol modulo $N$. Assuming the $L$-functions of primitive Dirichlet characters modulo $N$ have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any $c > 0$ may be taken.
Let $λ$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and } λ(a) = λ(b) = -1. $$ This conditionally answers an analogue of the binary Goldbach problem for the Liouville function, posed by Shusterman. The latter is a consequence of a quantitative lower bound on the frequency of sign patterns attained by $(λ(n),λ(N-n))$, for sufficiently large primes $N$. We show, assuming GRH, that there is a constant $C > 0$ such that for each pattern $(η_1,η_2) \in \{-1,+1\}^2$ and each prime $N \geq N_0$, $$ |\{n < N : (λ(n),λ(N-n)) = (η_1,η_2)\}| \gg N e^{-C(\log \log N)^{6}}. $$ The proof makes essential use of the Pierce expansion of rational numbers $n/N$, which may be of interest in other binary problems.
Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) = f(n+h) \neq 0\} $$ has logarithmic density 0. We also prove a converse result, along with an application to the Fourier coefficients of holomorphic cusp forms. The proof involves analysing the value distribution of $f$ using the compositions $|f|^{it}$, relying crucially on various applications of Tao's theorem on logarithmically-averaged correlations of non-pretentious multiplicative functions. Further key inputs arise from the inverse theory of sumsets in continuous additive combinatorics.
Let $λ$ denote the Liouville function. We show that for all sufficiently large integers $N$, the (non-trivial) convolution sum bound $$ \left|\sum_{1 \leq n < N} λ(n) λ(N-n)\right| < N-1 $$ holds. This (essentially) answers a question posed at the 2018 AIM workshop on Sarnak's conjecture.
Let $χ$ be a primitive character modulo $q$, and let $δ> 0$. Assuming that $χ$ has large order $d$, for any $d$th root of unity $α$ we obtain non-trivial upper bounds for the number of $n \leq x$ such that $χ(n) = α$, provided $x > q^δ$. This improves upon a previous result of the first author by removing restrictions on $q$ and $d$. As a corollary, we deduce that if the largest prime factor of $d$ satisfies $P^+(d) \to \infty$ then the level set $χ(n) = α$ has $o(x)$ such solutions whenever $x > q^δ$, for any fixed $δ> 0$. Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters $χ^\ell$, averaged over $1 \leq \ell \leq d-1$, due to the first author, which goes beyond Burgess' theorem as soon as $d$ is sufficiently large. We in fact show the alternative result that either (a) the partial sum of $χ$ itself, or (b) the partial sum of $χ^\ell$, for ``almost all'' $1 \leq \ell \leq d-1$, exhibits cancellation on the interval $[1,q^δ]$, for any fixed $δ> 0$. By an analogous method, we also show that the Pólya-Vinogradov inequality may be improved for either $χ$ itself or for almost all $χ^\ell$, with $1 \leq \ell \leq d-1$. In particular, our averaged estimates are non-trivial whenever $χ$ has sufficiently large even order $d$.
Let $χ$ be a primitive character modulo a prime $q$, and let $δ> 0$. It has previously been observed that if $χ$ has large order $d \geq d_0(δ)$ then $χ(n) \neq 1$ for some $n \leq q^δ$, in analogy with Vinogradov's conjecture on quadratic non-residues. We give a new and simple proof of this fact. We show, furthermore, that if $d$ is squarefree then for any $d$th root of unity $α$ the number of $n \leq x$ such that $χ(n) = α$ is $o_{d \to \infty}(x)$ whenever $x > q^δ$. Consequently, when $χ$ has sufficiently large order the sequence $(χ(n))_{n \leq q^δ}$ cannot cluster near $1$ for any $δ> 0$. Our proof relies on a second moment estimate for short sums of the characters $χ^\ell$, averaged over $1 \leq \ell \leq d-1$, that is non-trivial whenever $d$ has no small prime factors. In particular, given any $δ> 0$ we show that for all but $o(d)$ powers $1 \leq \ell \leq d-1$, the partial sums of $χ^\ell$ exhibit cancellation in intervals $n \leq q^δ$ as long as $d \geq d_0(δ)$ is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime $3 \leq d \leq q-1$, the Pólya-Vinogradov inequality may be improved for $χ^\ell$ on average over $1 \leq \ell \leq d-1$, extending work of Granville and Soundararajan.
Motivated by questions about the typical sizes of gaps $|f(n+1)-f(n)|$ in the sequence $(f(n))_n$, where $f$ is an integer-valued multiplicative function, we investigate the set of solutions $$ \{n \in \mathbb{N} : f(n+a) = f(n) + b\}, \quad ab \neq 0. $$ We formulate a conjecture classifying those multiplicative functions for which this set has logarithmic density zero, and prove that the conjectured classification is tight. Moreover, using techniques from additive combinatorics building on previous work of the author, we show how to reduce the classification problem to the study of "local power maps" modulo prime $\ell$, i.e., maps $g: \mathbb{N} \to \mathbb{Z}$ for which there is $0 \leq k_{\ell} < \ell-1$ such that $$ g(n) \equiv n^{k_{\ell}} \pmod{\ell} \text{ for all } n \in \mathbb{N}. $$ We prove a partial result towards our classification conjecture by employing a strategy of N. Jones that uses Kummer theory to study local power maps modulo many primes $\ell$.
We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative $f:\mathbb{F}_q[t]\to\{-1,+1\}$ with $q$ odd has bounded short interval sums if and only if $f$ coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over $\mathbb{Z}$ that such modified characters are extremal with respect to the growth rate of partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of $\mathbb{F}_{q}[t]$. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on $\mathbb{F}_q[t]$. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.
We study for bounded multiplicative functions $f$ sums of the form \begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n), \end{align*} establishing that their variance over residue classes $a \pmod q$ is small as soon as $q=o(x)$, for almost all moduli $q$, with a nearly power-saving exceptional set of $q$. This improves and generalizes previous results of Hooley on Barban-Davenport-Halberstam-type theorems for such $f$, and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well-known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli $q$ in the cases where $q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every $q$. These results are special cases of a "hybrid result" that works for sums of $f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki-Radziwill theorem on multiplicative functions in short intervals. We also consider the maximal deviation of $f$ over all residue classes $a\pmod q$ for $q\leq x^{1/2-\varepsilon}$, and show that it is small for "smooth-supported" $f$, again apart from a nearly power-saving set of exceptional $q$, thus providing a smaller exceptional set than what follows from Bombieri-Vinogradov-type theorems. As an application of our methods, we consider Linnik-type problems for products of exactly three primes, and in particular prove results relating to a ternary version of a conjecture of Erdős on representing every element of the multiplicative group $\mathbb{Z}_p^{\times}$ as the product of two primes less than $p$.
Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts $h_1,h_2$ the two-point correlation $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\overline{f}(n+h_2)}$$ tends to $0$ along a set of $x\in\mathbb{N}$ of full upper logarithmic density. We also show that the same result holds for the $k$-point correlations $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\cdots f(n+h_k)}$$ if $k$ is odd and $f$ is a real-valued non-pretentious function. Previously, the vanishing of correlations was known only under stronger non-pretentiousness hypotheses on $f$ by the works of Tao, and Tao and the third author. We derive several applications, including: (i) A classification of $\pm 1$-valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. (ii) A proof that a class of "Liouville-like" functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. (iii) Constructing examples of multiplicative $f:\mathbb{N}\to \{-1,0,1\}$ with a given (unique) Furstenberg system, answering a question of Lemańczyk. (iv) A density version of the Erdős discrepancy theorem of Tao.
We show that counts of squarefree integers up to $X$ in short intervals of size $H$ tend to a Gaussian distribution as long as $H\rightarrow\infty$ and $H = X^{o(1)}$. This answers a question posed by R.R. Hall in 1989. More generally we prove a variant of Donsker's theorem, showing that these counts scale to a fractional Brownian motion with Hurst parameter $1/4$. In fact we are able to prove these results hold in general for collections of $B$-free integers as long as the sieving set $B$ satisfies a very mild regularity property, for Hurst parameter varying with the set $B$.
We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)χ(G)ξ(G), \end{equation} where $χ$ is a Dirichlet character and $ξ$ is a short interval character over $\mathbb{F}_q[t].$ We then deduce versions of the Matomäki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where $q$ is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of $q$. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that $q$ is a power of $2$. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erdős discrepancy problem over $\mathbb{F}_q[t]$.
Let $f$ be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{λ_f(n)\}_n$ be its sequence of normalized Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of $K$, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{λ_f(n)\}_{n \in \mathcal{N}_K}$. More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $λ_f(n)λ_f(n') < 0$ where $n'$ is the first element of $\mathcal{N}_K$ greater than $n$ for which $λ_f(n') \neq 0$. For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 : m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,ε} X^{1/2-ε}$ sign changes for $λ_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$.
We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estimate for short character sums, and upper bounds for $L(1,χ)$ and $L(1+it,χ)$) are more-or-less "equivalent". We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.
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.