## Learning objectives

Undertanding the basic techniques of Number Theory (real and complex analysis)

## Course unit content

Basic Number Theory; distribution of prime numbers; arithmetical functions; sieve methods; the Riemann zeta-function and applications

## Full programme

Distribution of prime numbers: Chebyshev's theorems, Mertens's formulas, Selberg's formulas.

Elementary arithmetical functions: Multiplicative and totally multiplicative functions, Dirichlet product and the hyperbola method.

Sieve Methods: Sketch of Brun's combinatorial sieve and some applications.

The large sieve and its applications.

The Riemann zeta function and some properties, sketch of the analytic proof of the Prime Number Theorem.

Goldbach's problem: additive problems and the circle method.

## Bibliography

T. M. APOSTOL, Introduction to Analytic Number Theory, Springer, Berlino, 1975.

K. CHANDRASEKHARAN, Introduction to Analytic Number Theory, Springer, Berlino, 1968.

H. DAVENPORT, Multiplicative Number Theory, terza edizione, Springer, Berlino, 2001.

H. M. EDWARDS, Riemann's Zeta Function, Academic Press, 1974. Ristampa Dover, 2001.

G. H. HARDY & E. M. WRIGHT, An Introduction to the Theory of Numbers, quinta edizione, Oxford Science Publications, Oxford, 1979.

L. K. HUA, Introduction to Number Theory, Springer, Berlino, 1982.

E. LANDAU, Elementary Number Theory, Chelsea, New York, 1960.

H. L. MONTGOMERY & R. C. VAUGHAN, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge, 2006.

## Assessment methods and criteria

The student will deliver a 50 minute lecture on a topic chosen with the lecturer

## Other information

Lecture notes are available from the lecturer's own web page