ANALYTIC NUMBER THEORYMastermath  Fall 2016[estudiegids Bachelor wiskunde  eprospectus Master mathematics  Homepage Mathematisch Instituut ] 

Teachers: 
Dr. JanHendrik Evertse email evertse at math.leidenuniv.nl
Dr. Adelina Manzateanu
Assistant (responsible for grading the homework): 
Time/Place: 
Mondays 13:4515:30 Snellius 401
September 5December 12, 2016, with the exception of October 3, November 7 
EC points:  6 
Examination: 
The examination consists of four
homework assignments and a written exam in which you are asked questions
about the contents of the course and about the homework exercises.
Your grade for the exam, and the average of your grades of the four
homework assignments, both contribute 50% to your final grade
for Analytic number theory.
Possible resits will be given in the form of an oral exam. 
Written exam: 
We have scheduled a written exam on
Thursday January 26, 1417 h. (25 pm), room 407/409.
Possible resits will be given in the form of individual oral exams.
In the written exam or oral resit
we may ask questions about the homework exercises and variations thereof.

Old exams: 
Exam January 22, 2015
Part of the course given last fall was given by Efthymios Sofos, and so its contents are partly different from those of the analytic number theory course given two years ago. You may also expect a question on the material treated by Efthymios. 
Homework assignments (pdf): 
HOMEWORK GRADES
Homework assignment 1:
Due October 17
The deadlines for submitting your homework are strict. Please do not forget to write or type your name and student number on your homework. To simplify grading it is preferable that you submit your homework in latex. Homework that is not well readable will not be graded. You may either deliver your homework at the course, give it to Marc Paul, or submit an electronic version to him by email. In case you submit your homework by email, it should be by means of a single pdffile, and typed in latex. 
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 are not discussed in the course and they will not be examined,
but all theorems, corollaries etc. are assumed to be known.
Chapter 0: Notation and prerequisites

Remarks: 
This course will probably not be given in 2017/2018.
This course is recommended for a Master's thesis project in Number Theory. 
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 and complex function theory in Leiden will do);
Algebra: abelian groups (small subset of course Algebra 1 in Leiden). 
Literature: 
Recommended for further reading:
Recent survey paper on estimates for π(x) (number of primes up to x) and related issues.
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 nonnegative kth powers.The proofs are based on the circle method of Hardy and Littlewood. ISBN 0521605830
Classic book on the distribution of prime numbers. ISBN 0521397898
Sieve theory and applications to primes in arithmetic progressions.
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 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 
Useful websites: 
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.
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 through the institute's network.
An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.
Mathematical preprints archive; number theory preprints are categorized under NT. 
Contents (tentative, may be subject to change): 
The first part of this course will be 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.
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.
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, 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. Using this circle method, they gave the first estimates for G(k). Following the circle method, together with later refinements, it will be shown during our course that G(3)≤9. 
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