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| Teacher: |
Dr. Jan-Hendrik Evertse
office 248, tel. 071-5277148,
email evertse at math.leidenuniv.nl
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| Time/Place: |
TBA
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| EC points: |
6 |
| Prerequisites: |
Analysis: differential and integral calculus
of real functions in several variables, convergence of series,
(uniform) convergence of sequences of functions,
basics of complex analysis
(courses Analysis 1,2,3,4 in Leiden);
Algebra: abelian groups, rings (subset of courses Algebra 1,2 in
Leiden).
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Examination
and homework: |
The examination consists of four
homework assignments and an oral exam in which you are asked questions
on the theory treated during the course and the homework exercises.
Both the marks for the homework and the oral exam
count for 50% in the final mark.
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| Remarks: |
This course will not be given in 2013/2014.
This course is recommended for a Master's thesis project in Number Theory.
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| Literature: |
There will be lecture notes.
Recommended books for further reading:
H. Davenport, Multiplicative Number Theory (2nd edition),
Springer Verlag, Graduate Texts in Mathematics 74, 1980
This book discusses the properties of the Riemann zeta function,
as well as those of Dirichlet L-functions. Further it gives the proofs
of de la Vallée Poussin of the prime number theorem and the
prime number theorem for arithmetic progressions. Lastly, it treats
some sieve theory.
ISBN 3-540-90533-2
A.E. Ingham, The distribution of prime numbers,
Cambridge University Press, 1932 (reissued in 1990)
Classic book on the distribution of prime numbers.
ISBN 0-521-39789-8
H. Iwaniec, E. Kowalski, Analytic Number Theory,
American Mathematical Society Colloquium Publications, vol. 53,
American Mathematical Society 2004.
Analytic number theory bible, containing a lot of material.
The proofs are rather sketchy.
ISBN 0-8218-3633-1
G.J.O. Jameson, The Prime Number Theorem, London Mathematical Society
Student Texts 53, Cambridge University Press 2003.
This book gives both a proof of the Prime Number Theorem based on
complex analysis which is similar to the one we give during the course,
as well as an elementary proof not using complex analysis.
The book should be accessible to third year students.
ISBN 0-521-89110-8
S. Lang, Algebraic Number Theory, Addison-Wesley, 1970.
The third part contains analytic number theory related to algebraic number
theory, such as a proof of the functional equation of the Dedekind zeta
function for algebraic number fields
(this is a generalization of the Riemann zeta
function), a proof of the functional equation for L-series related
to Hecke characters (generalizations of Dirichlet characters),
a proof of the Prime Ideal Theorem
(a generalization of the Prime Number Theorem).
We will not discuss these topics
during our course, but it is important related material.
ISBN 0-201-04201-0
S. Lang, Complex Analysis (4th edition),
Springer Verlag, Graduate Texts in Mathematics 103, 1999
This book gives a comprehensive introduction to complex analysis.
It includes topics relevant for number theory, such as elliptic functions
and a simple proof of the Prime Number Theorem, due to Newman.
ISBN 0-387-98592-1
D.J. Newman, Analytic Number Theory, Springer Verlag, Gruduate Texts
in Mathematics 177, 1998.
This book gives an introduction to analytic number theory,
including a simple proof of the Prime Number Theorem,
and various other topics,
such as an asymptotic formula for the number of partitions,
Waring's problem about the representation of integers
by sums of k-th powers,
etc.
ISBN 0-387-98308-2
E.C. Titchmarsh, The theory of the Riemann zeta function
(2nd edition, revised by D.R. Heath-Brown), Oxford Science Publications,
Clarendon Press Oxford, 1986.
The title speaks for itself.
ISBN 0-19-853369-1
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Useful
websites:
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2010 Mathematics Subject Classification (MSC2010)
Official classification of mathematics subjects. The books in the Mathematical Institute library are classified according
to this classification.
The number theory books are classified under no. 11.
The Mathematical atlas, in particular the subcategories
Number Theory (MSC2010 11),
Zeta and L-functions: analytic theory (MSC2010 11M)
and
Multiplicative number theory (MSC2010 11N).
Number Theory Web
Website for the number theory community with many useful links.
Online number theory lecture notes
Long list of downloadable lecture notes on various branches of number theory including analytic number theory.
MathSciNet,
Zentralblatt
Online mathematical data bases which can be used to find abstracts of mathematical papers, lists of papers
of mathematicians, etc. MathSciNet covers the period 1940-... and Zentralblatt
1930-... .
These websites are accessible only through the institute's network.
MacTutor History
of Mathematics archive
An archive with all sorts of facts from the history of mathematics,
including biographies of the
most important mathematicians.
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| Contents: |
The focus of this course is on prime number theory.
In our course we give rather recent,
relatively simple proofs of both the Prime Number Theorem
and the Prime Number Theorem for arithmetic progressions, which are due to Newman.
If time permits,
we deduce the functional equation for the Riemann zeta function.
In the first part of the course we recall some complex analysis.
Below we give a short historical overview of the subject.
Let π(x) denote the number of primes not exceeding x. In 1798,
Legendre posed
the following conjecture:
π(x)∼x/log x as x→∞, that is,
limx→∞π(x)(log x/x)=1.
The first step in proving this conjecture was made by
Chebyshev, who proved in
1851-52 that if limx→∞
π(x)(log x/x) exists,
then the limit must be equal
to 1. However, he was not able to prove the existence of the limit.
The next important contribution was due to
Riemann. In his famous
memoir
from 1859 (his only paper on number theory), he proved several results
on the function ζ(s)=∑n≥1
n-s
(now known as the Riemann zeta function)
and stated several conjectures about this function.
Riemann viewed ζ(s) as a function in the complex variable s,
and showed that it has an analytic continuation
to C\{1}, with
a simple pole with residue 1 at s=1.
Further, Riemann showed that (the analytic continuation of) ζ(s)
satisfies a functional equation
which relates ζ(s) to ζ(1-s). It is an easy consequence
of this functional equation,
that ζ(s) has simple zeros at all even negative integers
-2,-4,-6,... and that all other zeros lie in the critical strip
consisting of all complex numbers s with real part between 0 and 1.
Riemann stated a still unproved famous conjecture,
now known as the Riemann Hypothesis (RH):
all zeros of ζ(s) in the critical
strip lie on the axis of symmetry of the functional equation, that is on the
line of all complex numbers with real part ½.
In his memoir,
Riemann mentioned several results, without proof or just with a sketch of a
proof,
relating the distribution of the
zeros of ζ(s) to the distribution of the prime numbers. These results
were proved completely in the 1890's by
Hadamard and von Mangoldt. Finally, in 1896,
Hadamard and
de la Vallée Poussin independently proved Legendre's conjecture
stated above, now known as the Prime Number Theorem. Their proofs were based on
complex analysis, applied to the Riemann zeta function. Later, several other proofs
of the Prime Number Theorem were given, all based on complex analysis,
until in 1948
Erdös and
Selberg independently published an "elementary proof"
of the Prime Number Theorem, avoiding complex analysis.
Dirichlet may be viewed as the founder of analytic number theory. In 1839-1842,
he showed that each arithmetic progression a,a+q,a+2q,a+3q,...
contains infinitely many prime numbers. In his proof he used properties of
so-called L-functions L(s,χ)=
∑n≥1 χ(n)n-s
where χ is a character modulo q.
As it turned out, L-functions have many properties similar to those of the
Riemann zeta function: for instance they have an analytic continuation to
C and they satisfy a
functional equation. Further, there is a
Generalized Riemann Hypothesis (GRH) which asserts that all zeros of
(the analytic continuation of) L(s,χ)
in the critical strip lie on the line of complex numbers with real part ½.
De la Vallée Poussin proved the Prime Number Theorem for arithmetic
progressions which is as follows:
let π(x;a,q) denote the number of primes not exceeding x
in the arithmetic progression a,a+q,a+2q,a+3q,... . Assume that 0<a<q
and that a is coprime with
q. Let φ(q) denote the number of positive integers smaller than q that are coprime with q.
Then π(x;a,q)∼φ(q)-1x/log x as x→∞.
This shows that roughly speaking, the primes are evenly distributed over
the prime residue classes modulo q.
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