*Course description:*
Our aim is to provide a classical introduction
to analytic number theory, focused on the connection between the zeros
of the Riemann z*(s)*-function and prime
numbers. We will follow the history and development of a beautiful discipline,
rich in problems, methods and ideas. The highlights are a proof of Dirichlet's
Theorem about the distribution of primes in arithmetic progressions, and
of the Prime Number Theorem, but the course is much more than just the
proof of two famous theorems.

*Topics:*

**Arithmetical functions**. Additive and multiplicative functions,
convolution, the Möbius inversion formulas, partial summation,
mean value of some functions, elementary prime number estimates.

**Dirichlet series.** The general theory of Dirichlet series, convergence,
Euler products, Perron's type formulas.

**Dirichlet's theorem.** The prime modulus case, characters, *L*--functions,
the general case, primitive characters and Gaussian sums. (*If time permits:
*quadratic forms, Dirichlet's class number formula.)

**Riemann's memoir.** The z*(s)*-function,
analytic properties of z*(s)* and *L*(s,c),
integral functions of order 1, the G--function,
the zeros of z*(s) *and *L*(s,c),
the Riemann Hypothesis (a XIX* ^{th}* century problem for the
XXI

**Other prime number sums.** (*If time permits: *The Goldbach--Vinogradov
theorem, the Polya--Vinogradov inequality, the large sieve, the Bombieri--Vinogradov
theorem.)