Università Roma Tre

Notations of analytic number theory. Euler's constant, the Dirichlet problem for the average number of divisors of an integer, the hyperbole method.

Chebichev theorems. Mertens theorem.

Dirichlet theorem first in arithmetic progression. The real function ζ (s), The Dirichlet theorem first in arithmetic progression, L-series of Dirichlet L (s, χ), productive.

Gaussian sums, Poisson sum formula, application to Gaussian sums. Dirichlet characters, determination of explicit characters,
orthogonality laws of characters, Dirichlet's theorem in the general case.

Analytical extension to s 0 of the function ζ (s) and of the Dirichlet L-series. Demonstration of de la Valleé Poussin that the L-series does not cancel each other out in s = 1. Mertens theorem first in arithmetic progression.

The Riemann function ζ. The Riemann article and the analytical extension of ζ (s). Riemann program for the proof of the prime number theorem. Riemann proof of the functional equation for ζ (s), trivial zeros for ζ (s). Hadamard products. Whole functions of finite order. Hadamard theorem for integer functions of order one. Distribution of the zeros of whole functions of order one. The ordinate of the non-trivial zero smaller than ζ is in absolute value 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 free of zeros for ζ (Hadamard theorem - de La Valleé Poussin 1896). Von Mangoldt's formula for N (T).

Distribution of the first. The explicit formula for the function ψ (x), the discontinuous integral of Perron. The prime number theorem. Consequences of the Riemann hypothesis.

Davenport, Harold, Multiplicative number theory. Graduate Texts in Mathematics, 74.Springer-Verlag, New York, (2000).

Tenenbaum, G erald, Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, (1995).