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 Translation missing. German original: Studienleistung: Erreichen von mindestens 50 Prozent der Punkte aus den wöchentlich zu bearbeitenden Übungsaufgaben. Prüfungsleistung: Klausur oder mündliche Prüfung 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

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Basismodulen, sowie in den Aufbaumodulen Funktionentheorie und Vektoranalysis sowie Zahlentheorie vermittelt werden.

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