Georg Friedrich Bernhard Riemann, 1826-1866,
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Introduction
This course is sufficiently
self contained that any course of elementary number theory or abstract
algebra
would be a sufficient prerequisite.
The primes are the basic building blocks for the
multiplicative
structure of the integers and their frequency and density is
fundamental.
One of the underlying motives for the
course
is an amazing and beautiful explicit formula, discovered by
Georg Friedrich Bernhard Riemann, 1826-1866,
connecting the prime numbers with the zeros of the Riemann zeta-function. The Riemann Hypothesis concerning the zeros is now the most important unsolved problem in mathematics. For an account of this and connected questions see the article by Enrico Bombieri at http://www.claymath.org/prizeproblems/riemann.htm. We will follow the very first tentative steps taken in this direction by Chebyshev, Mertens, Hadamard and de la Vallée Poussin.
Pafnuty Lvovich
Chebyshev Franz
Carl Joseph Mertens Jacques Salomon
Hadamard
1821-1894
1840
- 1927
1865 - 1963
Charles Jean Gustave NicolasBaron de la Vallée Poussin, 1866 - 1962
Closely interwoven with the Riemann zeta function
and
its generalisations are theta functions and modulur functions, first
investigated
in detail by Jacobi.
Other fundamental questions often
arise
from the study of the solubility of diophantine equations.
Sometimes
the solution of a diophantine equation can be made to depend on
approximations
to real numbers by rationals. An example is Pells' equation x²-dy²=1.
One
simple
theorem
on such approximations is due to Dirichlet and involves
the first use of the box principle. This also leads to Liouville's
theorem
Johann Peter Gustav Lejeune Dirichlet, 1805 - 1859 Joseph Liouville, 1809 - 1882
on the approximation of algebraic numbers and enables us to construct simple examples of transcendental numbers.
Godfrey
Harold Hardy, 1877-1947 Srinivasa
Ramanujan, 1887-1920
Hardy and Ramanujan found a remarkable
approximation to the partition function, and this was later refined by
Rademacher to give an exact formula.
Hans Rademacher, 1892-1969 John Edensor Littlewood, 1885-1977
The underlying ideas suggested to Hardy and Littlewood that
similar methods could be applied to Waring's problem and the Goldbach
problems. Waring
had asserted in 1770 that every positive number is the sum of
at most four squares, nine cubes, nineteen biquadrates "and so on
...". There is now an essentially complete proof of this
assertion and the core of the work is through refined versions of the
Hardy-Littlewood method.
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