Alexander P. Mangerel

activity

20152026

most citedMultiplicative functions that are close to their mean

13 citations · 45 across the 29 of their papers we have counts for

collaborators

33 papers

math.NT2026

On Sárközy's Local Extrema Conjectures

Alexander P. Mangerel

Let $f : N \to (0, \infty)$ be a multiplicative function, and let $ω > 0$ . We consider the frequency with which patterns of the form $$ f(n) > ω\max\{f(n-1),f(n+1)\},…

math.NT2026

On a conjecture of Goldmakher

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}…

math.NT2026

A Halász-type asymptotic formula for logarithmic means and its consequences

Oleksiy Klurman, Alexander P. Mangerel

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 app…

math.NT2025

Sharp bounds for maximal sums of odd order Dirichlet characters

Alexander P. Mangerel

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…

math.NT2025

On a rigidity property for quadratic Gauss sums

Alexander P. Mangerel

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…

math.NT2024

On Shusterman's Goldbach-type problem for sign patterns of the Liouville function

Alexander P. Mangerel

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 a…

math.NT2024

On a Goldbach-type problem for the Liouville function

Alexander P. Mangerel

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|…

math.NT2024

Large sums of high order characters II

Alexander P. Mangerel, Yichen You

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 numbe…

math.NT2023

Gap problems for integer-valued multiplicative functions

Alexander P. Mangerel

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…

math.NT2023

On Equal Consecutive Values of Multiplicative Functions

Alexander P. Mangerel

Let $f : N \to C$ be a multiplicative function for which $p : ∣ f (p) ∣ \neq = 1 \sum \frac{1}{p} = \infty.$ We show under this condition alone that for any in…

math.NT2023★ 2 cited

On Elliott's conjecture and applications

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

Let $f : N \to D$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we s…

math.NT2022

Large Sums of High Order Characters

Alexander P. Mangerel

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…

math.NT2022

Sign Changes of Fourier Coefficients of Cusp Forms at Norm Form Arguments

Alexander P. Mangerel

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 $K / Q$ is…

math.NT2022★ 1 cited

Beyond the Erdős discrepancy problem in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

We characterize the limiting behavior of partial sums of multiplicative functions $f : F_{q} [t] \to S^{1}$ . In contrast to the number field setting, the characterization depends…

math.NT2021

Three conjectures about character sums

Andrew Granville, Alexander P. Mangerel

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya-Vinogradov theorem for maxim…

math.NT2021

Squarefrees are Gaussian in short intervals

Ofir Gorodetsky, Alexander P. Mangerel, Brad Rodgers

We show that counts of squarefree integers up to $X$ in short intervals of size $H$ tend to a Gaussian distribution as long as $H \to \infty$ and $H = X^{o (1)}$ . This answers…

math.NT2021

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

Alexander P. Mangerel

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 add…

math.NT2021

Divisor-bounded multiplicative functions in short intervals

Alexander P. Mangerel

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. Ou…

math.NT2020

Monotone chains of Fourier coefficients of Hecke cusp forms

Oleksiy Klurman, Alexander Mangerel

We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_…

math.NT2020★ 4 cited

Correlations of multiplicative functions in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

We develop an approach to study character sums, weighted by a multiplicative function $f : F_{q} [t] \to S^{1}$ , of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)χ(G)ξ(G)…

math.NT2020★ 2 cited

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Alexander P. Mangerel

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 $X^{η}$ -sm…

math.NT2019★ 13 cited

Multiplicative functions that are close to their mean

Oleksiy Klurman, Alexander P. Mangerel, Cosmin Pohoata +1

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 resul…

math.NT2019★ 11 cited

Multiplicative functions in short arithmetic progressions

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

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 varianc…

math.NT2019

Short Character Sums and the Pólya-Vinogradov Inequality

Alexander P. Mangerel

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…