Università Roma Tre

1. notations of analytic number theory. Euler's constant, The Dirichlet problem for the average number of divisors of an integer, the Hyperbola method. Chebichev Theorems. Mertens Theorems.

2. Dirichlet Theorem for primes in arithmetic progression. The real function ζ(s), Dirichlet L-series L (s, χ), infinite products. Gauss sums, Poisson summation the formula , an application to Gauss sums. Dirichlet character explicit determination, proof of the orthogonality laws for characters, the Dirichlet theorem in the general case. Analytic extension at s 0 of the function ζ (s) and of the L-series of Dirichlet. de la Valle Poussin's proof that L-series do not vanish at s = 1 (L (1, χ) 0). Mertens theorem for primes in arithmetic progression.

3. The Riemann ζ function. The article of Riemann and the analytic extension of ζ (s). Riemann program for proof of the prime numbers Theorem. Proof of the functional equation for the Riemann ζ (s), trivial zeros for ζ (s). Hadamard products. Entire functions of finite order. Hadamard theorem for entire functions of order one. Distribution of zeros of entire functions of order one. The ordinate of the smallest zero non-trivial, in absolute value of ζ is greater than 6.5. Logarithmic derivatives of the function ξ (s). The series of reciprocals of zeros in the critical strip. The zeta function has no zeros on the line Re(s) = 1. The Gamma function. Region with no zeros for ζ (Hadamard - La Valle Poussin Theorem 1896). The formula of von Mangoldt for N(T).

4. The distribution of primes. The explicit formula for the function ψ(x), the discontinuous integral of Perron. Prime Number Theorem. Consequences of the Riemann hypothesis.

Harold Davenport. Multiplicative Number Theory. (Graduate Texts in Mathematics) Springer.
Gerald Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. (Cambridge Studies in Advanced Mathematics) CUP.
Tom M. Apostol. Introduction to Analytic Number Theory. (Undergraduate Texts in Mathematics) Springer.
Henryk Iwaniec, Emmanuel Kowalski. Analytic Number Theory (Colloquium Publications, Vol. 53) (Colloquium Publications (Amer Mathematical Soc)).
Paul T. Bateman, Harold G. Diamond. Analytic Number Theory: An Introductory Course