# [Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (07/04/2016, Efrat Bank, Alexander Mangerel)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Tue Apr 5 11:20:53 EDT 2016

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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 7 avril 2016 / Thursday, April 7, 2016

HEURE / TIME :
10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Efrat Bank (Michigan)

TITRE / TITLE :
Prime polynomial values of linear functions in short intervals.

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field.

A key role is played by the computation of some Galois groups.

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DATE :
Le jeudi 7 avril 2016 / Thursday, April 7, 2016

HEURE / TIME :
14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Alexander Mangerel (University of Toronto)

TITRE / TITLE :
On the Distribution of Integers with Restricted Prime Factors

LIEU / PLACE :
Concordia University, Library Building, 9th floor, Salle/Room 921-04

RESUME / ABSTRACT :
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 $\pi(x;\mathbf{E},\mathbf{k})$ 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 $x^{-1}\pi(x;\mathbf{E},\mathbf{k})$, 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 $\pi(x;\mathbf{E},\mathbf{k})$ which contradicts these heuristics in the case that $E_j(x)^2 \leq k_j \leq \log^{\frac{2}{3}-\epsilon} x$ for each $j$ under mild hypotheses. These results are achieved via an application of the saddle-point method. As a particular application, we prove an asymptotic formula for the number of integers with a fixed number of prime factors from specific arithmetic progressions, which generalizes a result due to Delange.

On the other hand, we show that a quasi-Poisson behaviour occurs uniformly in the range $E_j(x)^{\epsilon} \ll k_j \ll E_j(x)^{1-\epsilon}$, as well as a verification of this heuristic when each $k_j/E_j(x) = 1+o(1)$. This is done with an altogether different method, in particular through an strengthening and extension of effective mean value estimates due to Hal\’{a}sz.  We will also discuss other applications of this second method.

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Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
Andrew Granville (andrew at dms.umontreal.ca)
Dimitris Koukoulopoulos (koukoulo at dms.umontreal.ca)
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http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html