The Selberg Sieve - Lecture 1

Explore the foundations of the Selberg sieve in this comprehensive lecture from the Workshop on Additive Combinatorics. Delve into heuristics about twin primes, including naive approaches and the Hardy-Littlewood method. Examine probability concepts and learn about the goals for upcoming lectures. Investigate the simple Sieve of Eratosthenes and its extension to the Bateman-Horn Conjecture. Discover techniques for detecting primes in sequences and analyze examples. Study simple upper bound sieves and their application to twin primes. Conclude with an exploration of the Chinese remainder theorem and its implications for multiplicative functions in sieve theory.

Syllabus

The Selberg sieve Lecture 1
Introduction
Heuristics about the number of twin primes
Naive heuristics
Hardy - Little Wood
Probability
Goal of the next lectures: Bounds of the forms
Simple Sieve: Sieve of Eratosthenes
Extension Bateman - Horn Conjecture
Goal: detect primes in sequences
Example: n prime
Simple upper bound sieve
Twin primes
By Chinese remainder theorem wd is multiplicative