Searcharxiv⌕ Search

arXiv subjects

Dimitris Koukoulopoulos

Publications and source records attributed to Dimitris Koukoulopoulos.

30 records · Page 2Linked to original sources
arXiv2016

Sums of Euler products and statistics of elliptic curves

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang-Trotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

math.NT↗
arXiv2015

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

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 the natural generalization of this to $k$-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang-Maynard-Tao theorem.

math.NT↗
arXiv2015

Group structures of elliptic curves over finite fields

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\le M$ and $k\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\le (\log M)^{2-ε}$, then a density zero proportion of the groups in question actually arise as the group of points on some elliptic curve over some prime finite field. On the other hand, if $K\ge (\log M)^{2+ε}$, they conjectured that a density one proportion of the groups in question arise as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range $K\le (\log M)^{2-ε}$, and we prove that the second part of their conjecture holds in the limited range $K\ge M^{4+ε}$. In the wider range $K\ge M^2$, we show that a positive density of the groups in question actually occur.

math.NT↗
arXiv2015

Primes in short arithmetic progressions

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+ε}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod q$, provided that $1\le Q^2\le h/x^{1/15+ε}$.

math.NT↗
arXiv2015

When the sieve works

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length $x$ with primes including some in $(\sqrt{x},x]$, using methods motivated by additive combinatorics.

math.NT↗
arXiv2015

The frequency of elliptic curve groups over prime finite fields

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-ε}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.

math.NT↗
arXiv2014

On the concentration of certain additive functions

We study the concentration of the distribution of an additive function, when the sequence of prime values of $f$ decays fast and has good spacing properties. In particular, we prove a conjecture by Erdos and Katai on the concentration of $f(n)=\sum_{p|n}(\log p)^{-c}$ when $c>1$.

math.NT↗
arXiv2013

On multiplicative functions which are small on average

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n 2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $μ(n)n^{it}$ for some $t$.

math.NT↗
arXiv2013

Localized factorizations of integers

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and k\ge 2. This generalizes a result by K. Ford when k=1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k+1)-dimensional multiplication table as well as how many distinct sums of k+1 Farey fractions there are modulo 1.

math.NT↗
arXiv2013

On the number of integers in a generalized multiplication table

Motivated by the Erdos multiplication table problem we study the following question: Given numbers N_1,...,N_{k+1}, how many distinct products of the form n_1...n_{k+1} with n_i 1. In the present paper we generalize these results by establishing the order of magnitude of A_{k+1}(N_1,...,N_{k+1}) for arbitrary choices of N_1,...,N_{k+1} when k is 2,3,4 or 5. Moreover, we obtain a partial answer to our question when k>5. Lastly, we develop a heuristic argument which explains why the limitation of our method is k=5 in general and we suggest ways of improving the results of this paper.

math.NT↗
arXiv2010

Divisors of shifted primes

We bound from below the number of shifted primes p+s<x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants.

math.NT↗