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German original

Analytic Number Theory
(dt. Analytische Zahlentheorie)

Level, degree of commitment in original study programme Advanced module, compulsory elective module
Forms of teaching and learning,
workload		 Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements		 9 CP
Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading		 German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency		 One semester,
Regelmäßig im Wechsel mit anderen advanced moduleen in Algebra oder Analysis
Person in charge of the module's outline Prof. Dr. Pablo Ramacher

Contents

  • Arithmetic functions and Dirichlet series,
  • Characters and summation formulas,
  • L-functions and Riemann's zeta-function,
  • Exponential sums and Dirichlet polynomials,
  • Sieve methods and applications of the Large Sieve,
  • Equidistribution results for prime numbers in residual classes,
  • holomorphic automorphic functions.

Qualification Goals

The students shall

  • learn to transfer, develop and apply analytic methods to number theoretical questions,
  • train analytical ways of thinking and working,
  • learn modern techniques for scientific work in this field,
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Complex Analysis and Vector Analysis, Number Theory.


Recommended Reading

  • Brüdern, J.: Einführung in die analytische Zahlentheorie, Springer.
  • Davenport, H.: Multiplicative Number Theory, Springer.
  • Iwaniec, H.: Analytic number theory, AMS Colloquium Publications.



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2021/22. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.