(Thesday 28 February) Introduction of the course. The notion of Dirichlet Character
(Tuesday 6 March) Constructions and examples of Dirichlet Characters
(Thursday 8 March) L-functions, the Riemann zeta function, Euler product, continuation to σ>0,
(Friday 9 March) Proof of the Dirichlet Theorem for primes in arithmetic progressions (first part), linear
combination of logarithm of L-functions and ortogonality relations.
(Tuesday 13 March) Gauss sums. Last step of the Proof of Dirichlet Theorem for primes in arithmetic progressions. The Dirichlet Hyperbola method.
(Thursday 15 March) Introduction to Complex Analysis: the notion of holomorhic function
(Friday 16 March) Introduction to Complex Analysis: power series of holomorphic functions
(Tuesday 20 March) Introduction to Complex Analysis: complex integration
(Friday 23 March) Exercises of complex analysis (Andam Mustafa)
(Tuesday 27 March) Application of complex analysis to the proof that L(1,χ)≠0 when χ is a real Dirichlet Character, The Riemann memoire.
(Thursday 29 March) Estensione meromorfa per ζ, Gli zeri banali per ζ
e simmetria degli zeri.
(Thursday 5 April) [Filippo Tolli] entire functions of finite order, characterization of entire functions of finite order without zeros. Jensen's Formula and consequences.
(Friday 6 April) [Filippo Tolli]
Weierstrass product in the case when \sum 1/|zn| (where zn are the zeroes) is convergent.
Hadamard factorization Theorem for entire funztions of order 1. Example of the factorization of sin(πz). Exercises and notes.
(Tuesday 10 April) The function ξ is entire of order 1. The infinite product for ξ. B = (log(4π)-γ-2)/2.
(Thursday 12 April) Sum of reciprocals of the zeros. The zeroes ρ= β+iγ satisfy |γ|>6.
(Friday 13 April) the ζ(1+it)≠0 for all t. The zero free region (beginning)