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Material for the written exam: We may ask questions about the following: Jan-Hendrik's part: Chapters of the lecture notes, both theory and exercises, with the following exceptions: Sections 3. We may ask you to apply the functional equations of the zeta-function or L-functions, but you don't have to study their proofs. Further, you do not have to know the proofs of the Tauberian theorems but you have to know them and be able to apply them.

Damaris' part: Everything that has been discussed during Damaris' classes have a look at the videos if you missed some of them , and the seven exercise sheets. More precisely, the following topics can be examined: sieving for squre-free values of polynomials; Brun's sieve; the large sieve inequality and its applications to sieving; Gauss sums the material from Section 3.

At the written exam you may expect questions of the following type: - prove a theorem or result from the lecture notes or from Damaris' classes as long as the proof is shorter than one page, say; of course we will not ask you to reproduce long proofs ; - applications of the theorems from the course; - solving an exercise such as in the lecture notes or the exercise sheets or a variation thereof.

Math Introduction to Analytic Number Theory (Fall )

Computation of the grades: Since the amount of homework in the first three assignments given by Jan-Hendrik was somewhat larger than the amount of the last four assignments given by Damaris, the first three assignments will get a heavier weight. Old exams: Exam covers only the material of Jan-Hendrik's part Exam ; covers Jan-Hendrik's part and Efthymios Sofos' part about the circle method Damaris' part has not been taught before so there are no old exam exercises about that. Analysis: differential and integral calculus of real functions in several variables, convergence of series, uniform convergence of sequences of functions, basics of complex analysis; 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 mentioned in Chapter 0.

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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. 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 L-functions and properties thereof, in particular that the analytic continuations of the L-functions 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 In the second part we will give an introduction to sieve methods, including the fundamental lemma of sieve theory, Selberg's method, bilinear form methods, and the Large Sieve inequality. If time permits, we will discuss in more depth primes in arithmetic progressions and the Bombieri-Vinogradov Theorem, as well as applications to the ternary Goldbach problem representation of odd integers as a sum of three primes.

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. Practical matters: Do not forget to write very well readable or type your name, university and student number on your homework.

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You may either submit your homework at the course, or electronically by email to both assistants. To simplify grading we prefer very much that you submit the pdf of a la tex-file.

Handwritten homework will be graded only if it is very well readable and has no erasures. Scanned handwritten homework submitted by email will be accepted only if it is by means of a single pdf-file. Photographs of individual pages will not be accepted. Assignments: exercises are taken from the lecture notes; in parentheses the maximal number of points for an exercise Assignment 1, Due October 4 1. Assignment 5, Due November 29 Exercise 2 of the second exercise sheet.

Assignment 6, Due December 6 Exercise 4 of the third exercise sheet exercise 4 has been modified on November Assignment 7, Due December 13 Exercise 1 of the fourth exercise sheet.

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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 etc. Below you find an overview of the chapters that are taught, together with the sheets with the exercises for the exercise classes. This course is recommended for a Master's thesis project in Number Theory.

Recommended for further reading: H. Lastly, it treats some sieve theory. ISBN H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, Cambridge University Press, , reissued in in the Cambridge Mathematical Library series. In this book, Davenport considers various classes of Diophantine equations and inequalities to be solved in integers. The proofs are based on the circle method of Hardy and Littlewood. ISBN J. Or you may charge the purchase through the ABE.

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