School of Mathematics and Statistics |
Department of Mathematics and Statistics |

**NUMBER THEORY SEMINAR**

**Fall 2007**

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).

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.

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 .

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.

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.

** PAST SEMINARS**

** 2006-2007**

** 2005-2006**

** 2004-2005**

Nathan Ng

nng362 at science.uottawa.ca

Tel. 613-562-5800 ext 3515

or

Damien Roy

droy at uottawa.ca

Tel. 613-562-5800 ext 3504