ANALYTIC NUMBER THEORYMastermath course  Fall 2020  8EC[ Mastermath schedule  Mastermath webpage  Homework assignments  Course notes ] 
Schedule: 
Classes: Thursdays September 10December 17, 14:0016:45 (24:45 pm)
Classes will be given online; details will follow.
Exam: Thursday January 21, 2021, 14:0017:00 (written, onsite or online)

Teachers and assistants: 
Teachers:
Dr. JanHendrik Evertse (Universiteit Leiden)
Dr. Adelina Manzateanu (Universiteit Leiden)
Assistants (responsible for the exercise classes and the grading of the homework): not yet known 
Examination: 
The examination will consist of some homework assignments and a written exam.

Old exams: 
Exam 20142015 (covers only the material on prime number theory)
Exam 20162017 (covers everything) 
Prerequisites: 
Analysis: differential and integral calculus
of real functions in several variables, convergence of series,
(uniform) convergence of sequences of functions.
It will be useful to have some knowledge of the basics of complex analysis. Everything which is needed from complex analysis is contained in Chapter 0.
Algebra: elementary group theory, mostly only about abelian groups. Chapter 0 of the course notes (see below) gives an overview of what will be used during the course. We will not discuss the contents of Chapter 0 and they will not be examined, but you are supposed to be familiar with the theorems and concepts discussed in Chapter 0. 
Course description: 
The first part of the course is about prime number theory.
Our ultimate goal is to prove the prime number theorem,
and more generally, the prime number theorem for arithmetic progressions.
The prime number theorem, proved by Hadamard and de la Valleé Poussin
in 1896, asserts that if π(x) denotes the number of primes
up to x, then
π(x)∼x/log x as x→∞, that is,
lim_{x→∞}π(x)(log x/x)=1.
The prime number theory for arithmetic progressions, proved by
de la Valleé Poussin in 1899, can be stated as follows.
Let a,q be integers such that 0<a<q
and a is coprime with q and let
π(x;a,q) denote the number of primes not exceeding x
in the arithmetic progression a,a+q,a+2q,a+3q,... .
Let φ(q) denote the number of positive integers smaller than q that are coprime with q. Then π(x;a,q)∼φ(q)^{1}x/log x as x→∞.
This shows that roughly speaking, the primes are evenly distributed over
the prime residue classes modulo q.
In the course we will start with elementary prime number theory and then discuss the necessary ingredients to prove the above results: Dirichlet series, Dirichlet characters, the Riemann zeta function and Lfunctions and properties thereof, in particular that the (analytic continuations of) the Lfunctions do not vanish on the line of complex numbers with real part equal to 1. We then prove the prime number theorem for arithmetic progressions by means of a relatively simple method based on complex analysis, developed by Newman around 1980. In the second part we will discuss Waring's problem, which is about whether for each integer k≥2 there is a number g such that every positive integer can be expressed as a sum of at most g kth powers. We intend to show that every sufficient large integer can be expressed as a sum of nine positive cubes. More precisely, we will give an asymptotic formula for the number of ways a given positive integer n can be expressed as a sum of nine positive cubes (i.e., with a main term and an error term of smaller order of magnitude). To achieve this we will employ the famous circle method, originating from Hardy, Ramanujan and Littlewood. 
Homework assignments:  Homework assignments and their deadlines of delivery will be posted here. Please note that these deadlines are strict. Assignments are posted about two weeks prior to the deadline. At least for the first part of the course, homework exercises will be selected from the exercises in the course notes. 
Course notes (pdf): 
In Chapter 0 we have collected some facts
from algebra and analysis that will be used in the course.
The contents of Chapter 0 will not be discussed in the course and they will not be examined, but all theorems, corollaries, concepts etc.
are assumed to be known. You may use this as a reference source during the course.
Chapter 0: Notation and prerequisites
This covers JanHendrik's part of the course. The notes covering Adelina's part will follow. 
Remarks: 
This course will probably not be given in 2021/2022.
This course is recommended for a Master's thesis project in Number Theory. 
Literature: 
Recommended for further reading:
This book discusses the properties of the Riemann zeta function, as well as those of Dirichlet Lfunctions. 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 3540905332
In this book, Davenport considers various classes of Diophantine equations and inequalities to be solved in integers. Under certain hypotheses he shows that these are solvable and obtains asymptotic formulas for the number of solutions whose coordinates have absolute values at most X, as X→∞. A particular instance of this is Waring's problem, that for every positive integer k≥3 there is g such that every positive integer can be expressed as a sum of g nonnegative kth powers.The proofs are based on the circle method of Hardy and Littlewood. ISBN 0521605830
Online survey paper on estimates for π(x) (number of primes up to x) and related issues.
Classic book on the distribution of prime numbers. ISBN 0521397898
Analytic number theory bible, containing a lot of material. The proofs are rather sketchy. ISBN 0821836331
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 0521891108
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 Lseries 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 0201042010
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 0387985921
This book discusses in detail the Riemann zeta function, Lfunctions, the Prime Number Theorem for arithmetic progressions and refinements thereof, and a brief introduction to sieve theory. ISBN10 0521849039
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 kth powers, etc. ISBN 0387983082
The title speaks for itself. ISBN 0198533691
This book discusses several applications of the HardyLittlewood circle method, such as Diophantine equations and inequalities, Waring's problem (like Davenport's book above, but with more recent refinements) and the ternary Goldbach problem (that every odd integer larger than 5 is the sum of three primes). ISBN 0521573475 It may also be interesting to have a look at Riemann's memoir from 1859 (you can download both the original German version and its English translation), in which he proved several results on the function ζ(s)=∑_{n≥1} n^{s} and among other things formulated the Riemann Hypothesis (in a different but equivalent form). 
Useful websites: 
Official classification of mathematics subjects. Number theory is classified under no. 11.
Website for the number theory community with many useful links.
Long list of downloadable lecture notes on various branches of number theory including analytic number theory.
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 to subscribers. If your department has subscribed to them, you may consult them through your department's network.
An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.
Freely accessible mathematical preprints archive; number theory preprints are categorized under NT. 
Historical overview: 
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,
lim_{x→∞}π(x)(log x/x)=1.
The first step in proving this conjecture was made by
Chebyshev, who proved in
185152 that if lim_{x→∞}
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.
First observe that ζ(s) converges for all complex numbers s with
real part larger than 1.
Riemann showed that ζ(s) has a unique analytic continuation
to ℂ\{1}
(that is, there exists a unique analytic function
on ℂ\{1} which coincides with ζ(s) on Re s >1),
with a simple pole with residue 1 at s=1.
Denoting this analytic continuation also by ζ(s),
Riemann showed that ζ(s)
satisfies a functional equation
which relates ζ(s) to ζ(1s). 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
Erdős 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 18391842, 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 socalled Lfunctions L(s,χ)= ∑_{n≥1} χ(n)n^{s} where χ is a character modulo q. As it turned out, Lfunctions 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)^{1}x/log x as x→∞. This shows that roughly speaking, the primes are evenly distributed over the prime residue classes modulo q. In 1770, Lagrange proved that every positive integer can be expressed as a sum of four squares. Subsequently, Waring claimed without proof that every positive integer is the sum of nine cubes, nineteen fourth powers, and so on. Denote by g(k) the smallest integer g such that every positive integer is expressable as a sum of g kth powers, and by G(k) the smallest G such that every sufficiently large positive integer is expressable as a sum of G kth powers, that is, all integers with the exception of at most finitely many can be expressed as a sum of G kth powers. So G(k)≤g(k) for all k. In 1909, Hilbert proved that g(k) is finite for all k. Building further on work of Hardy and Ramanujan, in the early 1920's, Hardy and Littlewood developed a new method in analytic number theory, their famous circle method. This method has been successfully applied to give asymptotic formulas for the number of solutions of various types of Diophantine equations, for instance, to give an asymptotic formula for the number of times that a given positive integer n can be represented as a sum of G kth powers, for G sufficiently large. Further, the circle method is the main tool in the solution of the ternary Goldbach problem (that every odd integer ≥7 is the sum of three primes), and it has been used to give a good asymptotic formula for the number of partitions of a given integer. 
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