# 66 KAM Mathematical Colloquium

## INTRODUCTION TO SIEVE METHODS

utery 24. dubna 2007 ve 14:00, poslucharna S3, treti patro

V patek 27. 4. 2007 v poslucharne S5 ve 14:00 prednese J. Friedlander druhou prednasku Introduction to Sieve Methods, tematicky uzce souvisejici, avsak nezavislou na prvni prednasce.

KAM MFF UK
Malostranske nam. 25
118 00 Praha 1

## Abstract

Many of the most famous old conjectures in mathematics arise from the theory of numbers and, of those, quite a few are concerned with prime numbers. One example is the Goldbach conjecture which predicts that every even integer, at least four, can be written as the sum of two primes.

Questions about the distribution of primes began to see important progress during the nineteenth century, for the most part due to new developments in harmonic analysis and the theory of functions of a complex variable. Unfortunately these analytic techniques seem not to be capable of being adapted to the counting of primes in any but the simplest subsequences of the integers, such as for example the primes occurring in a short interval.

Another line of attack, the ancient and elementary sieve method, seems more versatile and indeed can be formulated so as to provide an entry into rather general questions of this nature. Based on new ideas due to Brun and others during the first three quarters of the twentieth century, one was able to prove theorems which in various senses gave approximations to the desired conjectures. Whereas however, in contrast to the analytic method, one could say something worthwhile about a great many different problems, one could never seem (again in contrast to the analytic method) to prove the conjecture that was really wanted. Moreover, there were theoretical reasons which seemed to render this failure an inevitable outcome of the method.

Today, the most famous conjectures seem still to be out of reach. Nevertheless, during the past twenty years it has become possible, by modifying the sieve method and then combining it with data achieved by analytic methods, to prove results about prime numbers which had been inaccessible to either method alone.

These lectures will present a survey of some the older and the more recent developments of these topics. They can be attended independently. The second lecture does not presuppose the knowledge of the first lecture.