# MA3150 - Analytic Number Theory

Lessons are not given in the academic year 2023/2024

### Course content

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.

### Learning outcome

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.

### Learning methods and activities

Lectures and exercises. The lectures will be given in English if the course is attended by students who don't master a Scandinavian language. Students are free to choose Norwegian or English for the oral examination. The course will be given in spring in years of odd numbers.

### Further on evaluation

The student should present a given topic during the oral exam. In the case that the student receives an F/Fail as a final grade after both ordinary and re-sit exam, then the student must retake the course in its entirety. Submitted work that counts towards the final grade will also have to be retaken. For more information about grading and evaluation, see «Teaching methods and activities».

### Course materials

Will be announced at the start of the course.

More on the course
Facts

Version: 1
Credits:  7.5 SP
Study level: Second degree level

Coursework

No

Language of instruction: English, Norwegian

Location: Trondheim

Subject area(s)
• Mathematics
Contact information