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401-3100-68L  Introduction to Analytic Number Theory

SemesterAutumn Semester 2018
LecturersI. N. Petrow
Periodicitynon-recurring course
Language of instructionEnglish


AbstractThis 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.
ObjectiveThe 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<x is approximately (1/phi(q))*(x/log x), as x tends to infinity, where phi(q) is the Euler totient function.
ContentDeveloping the necessary techniques and theory to prove the prime number theorem in arithmetic progressions will lead us to the study of prime numbers by Chebyshev's method, to study techniques for summing arithmetic functions by Dirichlet series, multiplicative functions, L-series, characters of a finite abelian group, theory of integral functions, and a detailed study of the Riemann zeta function and Dirichlet's L-functions.
Lecture notesLecture notes will be provided for the course.
LiteratureMultiplicative Number Theory by Harold Davenport
Multiplicative Number Theory I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan
Analytic Number Theory by Henryk Iwaniec and Emmanuel Kowalski
Prerequisites / NoticeComplex analysis
Group theory
Linear algebra
Familiarity with the Fourier transform and Fourier series preferable but not required.