Analytic Number Theory

Analytic number theory studies the distribution of the prime numbers, based on methods from mathematical analysis. Of central importance is the study of the Riemann zeta function, which embodies both the additive and the multiplicative structure of the integers. It turns out that the localization of the zeros of this meromorphic function is closely related to the distribution of the primes. At the end of the nineteenth century, this insight led to the celebrated prime number theorem. The zeta function has been subject to intensive research ever since, but many fundamental questions remain open, of which the Riemann hypothesis undoubtedly is the most famous. Key words for the course: Arithmetic and multiplicative functions, Abel summation and Möbius inversion, Dirichlet series and Euler products, the Riemann zeta function, the functional equation for the zeta function, the gamma function, The Mellin transformation and Perron's formula, the prime number theorem, the Riemann hypothesis, Dirichlet characters, Dirichlet's theorem on primes in arithmetic progressions.

1. Knowledge: The student masters the basic concepts of analytic number theory, including selected arithmetic and multiplicative functions, Abel summation and Möbius inversion, the Mellin transformation and Perron's formula, Dirichlet series and Euler products, Dirichlet characters. The students knows both the additive and the multiplicative definition of the Riemanns zeta function, the functional equation and the basic of the zeta function and the gamma function. The student has an overview of and can formulate the central results and open problems of the subject, including the prime number theorem and the Riemann hypothesis.

2. Skills: The student masters the basic methods of analytic number theory, including Abel summation and Möbius inversion, as well as the calculus of residues related to Perron's formula. The student is able to read research papers within selected parts of analytic number theory.

Lectures and exercises. Students are free to choose Norwegian or English for the oral examination. The course will be given in spring in years of odd numbers.