# C3.8 Analytic Number Theory - Material for the year 2020-2021

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Basic ideas of complex analysis. Elementary number theory. Some familiarity with Fourier series will be helpful but not essential.

16 lectures

### Assessment type:

- Written Examination

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 $X/\log X$.

In addition to the highlights mentioned above, students will gain experience with different types of Fourier transform and with the use of complex analysis.

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.

Full printed notes will be provided for the course, including the non-examinable topics (marked with asterisks above). The following books are relevant to the course.

- G. H. Hardy and E. M. Wright,
*An introduction to the Theory of Numbers*(Sixth edition, OUP 2008). Chapters 16, 17, 18. - H. Davenport,
*Multiplicative number theory*(Third Edition, Springer Graduate texts 74), selected parts of the first half. - M. du Sautoy,
*Music of the primes*(this is a popular book which could be useful background reading for the course).