General Prerequisites: Basic ideas of complex analysis. Elementary number theory. Some familiarity with Fourier series will be helpful but not essential.
Course Overview: The aim of this course is to study the prime numbers using the famous Riemann \(\zeta\)-function. In particular, we will study the connection between the primes and the zeros of the \(\zeta\)-function. We will state the Riemann hypothesis, perhaps the most famous unsolved problem in mathematics, and examine its implication for the distribution of primes. We will prove the prime number theorem, which states that the number of primes less than \(X\) is asymptotic to \(X\log X\).
Learning Outcomes: In addition to the highlights mentioned above, students will gain experience with different types of Fourier transform and with the use of complex analysis.
Course Synopsis: Introductory material on primes. Arithmetic functions: Möbius function, Euler's \(\phi\)-function, the divisor function, the \(\sigma\)-function. Multiplicativity. Dirichlet series and Euler products. The von Mangoldt function.
The Riemann \(\zeta\)-function for \(Re (s) > 1\). Euler's proof of the infinitude of primes. \(\zeta\) and the von Mongoldt function.
Schwarz functions on \(\mathbf{R}\), \(\mathbf{Z}\), \(\mathbf{R}/\mathbf{Z}\) and their Fourier transforms. *Inversion formulas and uniqueness*. The Poisson summation formula. The meromorphic continuation and functional equation of the \(\zeta\)-function. Poles and zeros of \(\zeta\) and statement of the Riemann hypothesis. Basic estimates for \(\zeta\).
The classical zero-free region. Proof of the prime number theorem. Implications of the Riemann hypothesis for the distribution of primes.
