This course considers the classical topics in analytic number theory, with a focus on tools coming from Fourier analysis. The list of topics to be covered is as follows:
1. Review of the basic elements of Fourier analysis: Fourier transform in L^1 and L^2; Plancherel's theorem; Tempered distributions; Fourier series; Convolution and approximations of the identity.
2. Diophantine approximations; Equidistribution of sequences; Notions of discrepancy; Erdös-Turán inequality; Irregularities of distribution.
3. Arithmetic functions; The Riemann zeta-function; Dirichlet characters; Dirichlet L-functions; Gauss sums; Primes in arithmetic progression; Functional equation for L-functions; Zero-free regions for zeta and other L-functions; Prime Number Theorem.
4. Consequences of the Riemann hypothesis; Explicit formulas; Pair correlation of zeros; Prime gaps; Extremal functions and Fourier optimization methods.
5. Geometry of numbers; Minkowski's convex body theorem.
References books:
1. H. Davenport, Multiplicative Number Theory, Third edition, Springer 2000.
2. H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Volume 53, 2004.
3. E. C. Titchmarsh, The theory of the Riemann zeta-function, Second Edition, Oxford Science Publications, 1986.
4. H. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics 97. Cambridge University Press, 2006.