AN INTRODUCTION TO SIEVE METHODS AND THEIR APPLICATIONS

By I. RADAK

CONTENTS SOME BASIC NOTIONS THE BIG ‘O’ AND LITTLE ‘O’ NOTATION THE M¨OBIUS FUNCTION THE TECHNIQUE OF PARTIAL SUMMATION CHEBYCHEFF’S THEOREM EXERCISES SOME ELEMENTARY SIEVES GENERALITIES THE LARGER SIEVE THE SQUARE SIEVE SIEVING USING DIRICHLET SERIES THE NORMAL ORDER METHOD A THEOREM OF HARDY AND RAMANUJAN THE NORMAL NUMBER OF PRIME DIVISORS OF A POLYNOMIAL PRIME ESTIMATES APPLICATION OF THE METHOD TO OTHER SEQUENCES THE TURÁN SIEVE THE BASIC INEQUALITY COUNTING IRREDUCIBLE POLYNOMIALS IN PX COUNTING IRREDUCIBLE POLYNOMIALS IN X SQUARE VALUES OF POLYNOMIALS AN APPLICATION WITH HILBERT SYMBOLS THE SIEVE OF ERATOSTHENES MERTENS’ THEOREM RANKIN’S TRICK AND THE FUNCTION XZ THE GENERAL SIEVE OF ERATOSTHENES AND APPLICATIONS BRUN’S SIEVE BRUN’S PURE SIEVE BRUN’S MAIN THEOREM SCHNIRELMAN’S THEOREM A THEOREM OF ROMANOFF SELBERG’S SIEVE CHEBYCHEFF’S THEOREM REVISITED THE BRUN–TITCHMARSH THEOREM AND APPLICATIONS THE LARGE SIEVE THE LARGE SIEVE INEQUALITY WEIGHTED SUMS OF DIRICHLET CHARACTERS AN AVERAGE RESULT THE BOMBIERI–VINOGRADOV THEOREM A GENERAL THEOREM THE TITCHMARSH DIVISOR PROBLEM THE LOWER BOUND SIEVE 10 1 THE LOWER BOUND SIEVE 10 2 TWIN PRIMES 10 3 QUANTITATIVE RESULTS AND VARIATIONS 10 4 APPLICATION TO PRIMITIVE ROOTS 10 5 EXERCISES NEW DIRECTIONS IN SIEVE THEORY 11 1 A DUALITY PRINCIPLE 11 2 A GENERAL FORMALISM 11 3 LINNIK’S PROBLEM FOR ELLIPTIC CURVES 11 4 LINNIK’S PROBLEM FOR CUSP FORMS 11 5 THE LARGE SIEVE INEQUALITY ON GLN 11 6 EXERCISES REFERENCES