Number Theory Seminar

Carleton University School of Mathematics and Statistics	University of Ottawa Department of Mathematics and Statistics

Time and Place: Monday, Oct. 22, 3pm-4pm, room KED B-015 Ottawa

Speaker: Arnaud Chadozeau (CRM, Université de Montréal)

Title: Distribution of numbers coprime to a given integer in small intervals

Abstract:

Form work of Montgomery and Vaughan, the number of integers coprime
to a given modulus q in an interval of length h is known to have
Gaussian distribution of mean and variance equal to h phi(q)/q,
provided h is suitably large. (Here, phi is the Euler totient function.)
We refine this statement for moduli q free of small prime factors (less than h).

Time and Place: Monday, Nov. 5, 3pm-4pm, room KED B-015 Ottawa

Speaker: Gary Walsh (University of Ottawa)

Title: Arithmetical properties of a sequence of integers related to a family of simultaneous Pell equations

Abstract:

We discuss necessary conditions for the existence of an integral solution to a
system of Pell equations, and some arithmetical results pertaining to the coefficients
of such solvable systems, improving upon recent work of Zhenfu Cao and his colleagues.

Time and Place: Monday, Nov. 12, 3pm-4pm, room KED B-015 Ottawa

Speaker: Youness Lamzouri (Université de Montréal)

Title: The two dimensional distribution of values of the Riemann zeta function on the one-line

Abstract:

In this talk we present several results on the joint distribution function of the argument
and the norm of the Riemann zeta function on the one line. Similar results for Dirichlet
L-functions at one are also given.

Here is the longer version of the abstract .

Time and Place: Monday, Nov. 26, 3pm-4pm, room KED B-015 Ottawa

Speaker: Shanta Laishram (University of Waterloo)

Title: Squares in Arithmetic Progression

Abstract:

A result of Fermat states that there are no four squares in an arithmetic
progression and Euler gave a general result that product of four terms of an
arithmetic progression is never a square. Hirata-Kohno, Laishram, Shorey
and Tijdeman extended Euler's result upto $109$ terms. For this, we
consider the Diophantine equation n(n+d)...(n+(k-1)d)=y^2
with n>= 1, d>= 2, k >= 3 and gcd(n, d)=1. In this talk, I
will give some history and discuss the above result and related results. In fact, in
a joint work with Shorey, we show that the above equation has no solution when
d <= 10^{10} or d has at most five prime divisors.

Time and Place: Monday, Dec. 3, room KED B-015 Ottawa

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: Diophantine equations and the generalized Riemann hypothesis
Abstract: We show how methods from the negative solution to Hilbert's tenth problem
may be used to show certain statements are equivalent to the unsolvability
of a Diophantine equation. When this happens, we will say the statement is
Diophantine. We show that the generalized Riemann hypothesis for a number
field is Diophantine. We also show the statement 'the generalized Riemann
hypothesis holds for every number field' is Diophantine. That is, there is
a Diophantine equation which has no solutions if and only if the
generalized Riemann hypothesis holds for every number field.

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