School of Mathematics and Statistics
Department of Mathematics and Statistics
NUMBER THEORY SEMINAR
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