# [Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (14/09/2017, Xianchang Meng Lior Bary-Soroker)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Tue Sep 12 09:27:01 EDT 2017

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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 14 septembre 2017 / Thursday, September 14, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Xianchang Meng (McGill)

TITRE / TITLE :
Chebyschev Bias

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
Chebyshev observed that there seems to be more prime numbers congruent to 3 \bmod 4 than that congruent to 1 \bmod 4. Rubinstein and Sarnak studied this problem using GRH and LI. Later Ford and Sneed studied the distribution of products of two primes in different arithmetic progressions. We would like to generalize this phenomenon to products of any of k \geq 3 primes in different arithmetic progressions. For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$  or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases. In our proof, we propose a different approach to solve this problem. If time permits, I would like to introduce some applications of this method to related problems.

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DATE :
Le jeudi 14 septembre 2017 / Thursday, September 14, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Lior Bary-Soroker (Tel-Aviv  University)

TITRE / TITLE :
Is a polynomial with plus minus 1 coefficient irreducible over the integers?

LIEU / PLACE :
Concordia University, Library Building, 9th floor, room LB 921-4

RESUME / ABSTRACT :
It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and that it has a big Galois group. For a few dozen of years, people have been interested whether the same holds when one considers sparse families of polynomials — notably, polynomials with plus-minus 1 coefficients. In particular, some guy on the street conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity. In this talk, I will discuss these types of problems, some approaches to attack them, and I will present some new results toward it, joint with Gady Kozma.

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Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)