#
Number Theory Research Group

Gauss famously described Mathematics as the "Queen of the Sciences"
and Number Theory as the "Queen of Mathematics". Humanity has probably
been fascinated by numbers since the discovery of counting. The
ancient Greeks established it as a subject (which they called
arithmetic): Books 7--10 of Euclid's Elements are devoted to number
theory. These books contain a treatment of the Euclidean algorithm, a
proof that there are infinitely many primes, a theory of
incommensurables (or irrationals -- the original proof that the square
root of 2 is irrational is attributed to Pythagoras). The sieve of
Eratosthenes is the basis of the modern sieve and Diophantus of
Alexandria began the systematic study of equations with integer
coefficients. The subject has flourished since the Renaissance and
today is as vigorous as ever: the proof of Fermat's Last Theorem
sought for over 300 years was solved a few years ago.
Until the middle of the 20th century, Number Theory was considered to
be the purest area of Mathematics and least likely to be sullied by
applications. This has all changed over the last 50 years with the
advent of digital computing, the growth of cryptography and the
emergence of links with dynamical systems and physical phenomena. Now
the subject is an exciting mix of ideas pursued for their own interest
and a rather exotic variety of applications.

People in the Number Theory Group at York
have varied interests, which include analytic number theory,
Diophantine approximation, geometry of numbers and connections with
dynamical systems. As well as participating in the departmental GANT
seminars , members of the group enjoy contacts with
mathematicians from all over the world.
Within the department, we overlap with the algebra
group, the analysis group and the networks and nonlinear dynamics
group.

We have close links with Edinburgh, Liverpool, Southampton, Queen
Mary and Westfield College, London and Warwick; and abroad with
Brisbane, Maynooth, Ostrava, Strasbourg and through an INTAS
programme, with Minsk, Marburg and St. Petersburg.

Current research areas:

** Analytic number theory** Concerned
with the distribution of prime numbers, the Riemann zeta function and
Diophantine equations, it owes its name from the techniques it uses
being drawn from analysis. Currently Richard Hall is studying the
distribution of the zeros of the Riemann zeta function.

**Diophantine approximation** A
quantitative and more general study of the qualitative fact that the
rationals are dense in the reals. We are interested in the metrical
theory, in which exceptional sets of points of measure zero for which
Diophantine inequalities do * not* hold are ignored -- this has
the advantage of often giving results of a striking
simplicity. Analytic concepts play an equal role with ideas from
measure and ergodic theory. Detta Dickinson (Maynooth), Maurice
Dodson and Simon Kristensen (Edinburgh) are investigating
applications to the theory of dynamical systems through the
phenomenom of `small divisors'. Deeper asymptotic formulae results
are being investigated with Jason Levesley and with Victor
Beresnevich (an EPSRC RA) and Sanju Velani (a RSUF), who have both
recently joined the department. They are also looking at how well
points in various sets of number theoretic and geometric interest can
be approximated. A major goal is to establish a `non-linear' metrical
theory, in which the variables lie on manifolds; Bernik (Minsk) and
his co-workers are also involved in this programme. General measure
theory aspects of the theory are being considered with Yann Bugeaud
(Strasbourg).

** Geometry of Numbers ** Its origins lie in
Minkowski's observation at the end of the 19th century that some
easily accessible geometric results having interesting consequences in
number theory. A typical problem is to find the circumstances under
which a given *n*-dimensional body contains a non-zero lattice
point. For some kinds of bodies this problem corresponds to questions
about the values of quadratic forms. Terence Jackson is studying
representation questions for definite forms and lattice constants of
bodies associated with indefinite forms.

** Ergodic theory and dynamical systems**
Number theoretic ideas have applications
in ergodic theory and dynamical systems and vice-versa. Recently
Zaq Coehlo and Bill Parry (Warwick) have found
connections between *p*-adic multiplication and Fibonacci
numbers.

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*Revised 14
April 2004*