Dimitris Koukoulopoulos

activity

20082026

most citedPretentious multiplicative functions and the prime number theorem for arithmetic progressions

25 citations · 200 across the 28 of their papers we have counts for

collaborators

30 papers

math.NT2026

A logarithmic structure theorem for multiplicative functions with small partial sums

Dimitris Koukoulopoulos

Let $D \in N$ , let $A > D + 1$ , and let $Q ⩾ 3$ . Consider the class of multiplicative functions $f : N \to C$ such that $|\sum_{n\leqslant x}f(n)|\le x(\log…

math.NT2025

Irreducibility and Galois groups of random reciprocal polynomials of large degree

David Hokken, Dimitris Koukoulopoulos

Let $A = a_{0} T^{m} + \sum_{j = 1}^{m - 1} a_{j} (T^{m - j} + T^{m + j}) + T^{2 m} + 1 \in Z [T]$ be a monic reciprocal polynomial of degree $2 m$ sampled randomly by selecting its coefficient…

math.NT2025

Erdős's integer dilation approximation problem and GCD graphs

Dimitris Koukoulopoulos, Youness Lamzouri, Jared Duker Lichtman

Let $A \subset R_{⩾ 1}$ be a countable set such that $lim sup_{x \to \infty} \frac{1}{l o g x} \sum_{α \in A \cap [1, x]} \frac{1}{α} > 0$ . We prove that, for ev…

math.NT2024

On Arratia's coupling and the Dirichlet law for the factors of a random integer

Tony Haddad, Dimitris Koukoulopoulos

Let $x \geq 2$ , let $N_{x}$ be an integer chosen uniformly at random from the set $Z \cap [1, x]$ , and let $(V_{1}, V_{2}, \dots)$ be a Poisson--Dirichlet process of parameter $1…

math.NT2024

An almost sharp quantitative version of the Duffin-Schaeffer conjecture

Dimitris Koukoulopoulos, James Maynard, Daodao Yang

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $ψ : N \to [0, 1/2]$ be a function such that the series $\sum…

math.NT2023★ 1 cited

A lower bound on the mean value of the Erdős-Hooley Delta function

Kevin Ford, Dimitris Koukoulopoulos, Terence Tao

We give an improved lower bound for the average of the Erdős-Hooley function $Δ (n)$ , namely $\sum_{n \leq x} Δ (n) ≫_{ε} x (lo g lo g x)^{1 + η - ε}$ for all $x\geqslant…

math.NT2023★ 5 cited

An upper bound on the mean value of the Erdős-Hooley Delta function

Dimitris Koukoulopoulos, Terence Tao

The Erdős-Hooley Delta function is defined for $n \in N$ as $Δ (n) = sup_{u \in R} # {d ∣ n : e^{u} < d \leq e^{u + 1}}$ . We prove that $\sum_{n\le x} Δ(n) \ll x(\log\log x)^{1…

math.NT2022

On the $j$ -th smallest modulus of a covering system with distinct moduli

Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux

Covering systems were introduced by Erdős in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounde…

math.NT2021

Rational approximations of irrational numbers

Dimitris Koukoulopoulos

Given quantities $Δ_{1}, Δ_{2}, \dots ⩾ 0$ , a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational…

math.NT2020★ 20 cited

Irreducibility of random polynomials: general measures

Lior Bary-Soroker, Dimitris Koukoulopoulos, Gady Kozma

Let $μ$ be a probability measure on $Z$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]…

math.NT2020★ 3 cited

A note on the natural density of product sets

Sandro Bettin, Dimitris Koukoulopoulos, Carlo Sanna

Given two sets of natural numbers $A$ and $B$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\…

math.NT2019★ 2 cited

The structure of multiplicative functions with small partial sums

Dimitris Koukoulopoulos, K. Soundararajan

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…

math.NT2019

Equal sums in random sets and the concentration of divisors

Kevin Ford, Ben Green, Dimitris Koukoulopoulos

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erdős-Hooley $Δ$ -function by $Δ(n) := \max_t \# \{d | n, \log d \in [t,…

math.NT2019

On the Duffin-Schaeffer conjecture

Dimitris Koukoulopoulos, James Maynard

Let $ψ : N \to R_{\geq 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $A$ of real numbers $α$ for which t…

math.NT2017★ 12 cited

Beyond the LSD method for the partial sums of multiplicative functions

Andrew Granville, Dimitris Koukoulopoulos

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,…

math.NT2016

Sieve weights and their smoothings

Andrew Granville, Dimitris Koukoulopoulos, James Maynard

We obtain asymptotic formulas for the $2 k$ th moments of partially smoothed divisor sums of the Möbius function. When $2 k$ is small compared with $A$ , the level of smoothing, then t…

math.GR2016★ 12 cited

Permutations contained in transitive subgroups

Sean Eberhard, Kevin Ford, Dimitris Koukoulopoulos

In the first paper in this series we estimated the probability that a random permutation $π \in S_{n}$ has a fixed set of a given size. In this paper, we elaborate on the sam…

math.NT2015★ 11 cited

Sums of Euler products and statistics of elliptic curves

Chantal David, Dimitris Koukoulopoulos, Ethan Smith

We present several results related to statistics for elliptic curves over a finite field $F_{p}$ as corollaries of a general theorem about averages of Euler products that we…

math.NT2014★ 8 cited

Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao

Andrew Granville, Daniel M. Kane, Dimitris Koukoulopoulos +1

We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p, p + h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider t…

math.NT2014★ 25 cited

The frequency and the structure of large character sums

Jonathan Bober, Leo Goldmakher, Andrew Granville +1

Let $M (χ)$ denote the maximum of $∣ \sum_{n \leq N} χ (n) ∣$ for a given non-principal Dirichlet character $χ (mod q)$ , and let $N_{χ}$ denote a point at which the maximum is attained. In t…

math.NT2014★ 6 cited

The frequency of elliptic curve groups over prime finite fields

Vorrapan Chandee, Chantal David, Dimitris Koukoulopoulos +1

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $F_{p}$ , we study the frequency to which a given group $G$ arises as a group of…

math.NT2014★ 9 cited

Primes in short arithmetic progressions

Dimitris Koukoulopoulos

Let $x, h$ and $Q$ be three parameters. We show that, for most moduli $q \leq Q$ and for most positive real numbers $y \leq x$ , every reduced arithmetic progression $a mod q$ has approx…

math.NT2012★ 6 cited

Group structures of elliptic curves over finite fields

Vorrapan Chandee, Chantal David, Dimitris Koukoulopoulos +1

It is well-known that if $E$ is an elliptic curve over the finite field $F_{p}$ , then $E (F_{p}) ≃ Z / m Z \times Z / mk Z$ for some pos…

math.NT2012★ 10 cited

When the sieve works

Andrew Granville, Dimitris Koukoulopoulos, Kaisa Matomäki

We are interested in classifying those sets of primes $P$ such that when we sieve out the integers up to $x$ by the primes in $P^{c}$ we are left with roughly the…

A logarithmic structure theorem for multiplicative functions with small partial sums

Irreducibility and Galois groups of random reciprocal polynomials of large degree

Erdős's integer dilation approximation problem and GCD graphs

On Arratia's coupling and the Dirichlet law for the factors of a random integer

An almost sharp quantitative version of the Duffin-Schaeffer conjecture

A lower bound on the mean value of the Erdős-Hooley Delta function

An upper bound on the mean value of the Erdős-Hooley Delta function

On the j-th smallest modulus of a covering system with distinct moduli

Rational approximations of irrational numbers

Irreducibility of random polynomials: general measures

A note on the natural density of product sets

The structure of multiplicative functions with small partial sums

Equal sums in random sets and the concentration of divisors

On the Duffin-Schaeffer conjecture

Beyond the LSD method for the partial sums of multiplicative functions

Sieve weights and their smoothings

Permutations contained in transitive subgroups

Sums of Euler products and statistics of elliptic curves

Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao

The frequency and the structure of large character sums

The frequency of elliptic curve groups over prime finite fields

Primes in short arithmetic progressions

Group structures of elliptic curves over finite fields

When the sieve works

On the $j$ -th smallest modulus of a covering system with distinct moduli