# Search result: Catalogue data in Autumn Semester 2018

Number Title Type ECTS Hours Lecturers Mathematics Bachelor Electives Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic 401-3113-68L Exponential Sums over Finite Fields W 8 credits 4G E. Kowalski Abstract Exponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory. Objective The goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields. Content Examples of elementary exponential sumsThe Riemann Hypothesis for curves and its applicationsDefinition of trace functions over finite fieldsThe formalism of the Riemann Hypothesis of DeligneSelected applications Lecture notes Lectures notes from various sources will be provided Literature Kowalski, "Exponential sums over finite fields, I: elementary methods:Iwaniec-Kowalski, "Analytic number theory", chapter 11Fouvry, Kowalski and Michel, "Trace functions over finite fields and their applications" 401-3100-68L Introduction to Analytic Number Theory W 8 credits 4G I. N. Petrow Abstract This course is an introduction to classical multiplicative analytic number theory. The main object of study is the distribution of the prime numbers in the integers. We will study arithmetic functions and learn the basic tools for manipulating and calculating their averages. We will make use of generating series and tools from complex analysis. Objective The main goal for the course is to prove the prime number theorem in arithmetic progressions: If gcd(a,q)=1, then the number of primes p = a mod q with p
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