Area, Lattice Points, and Exponential Sums

Transcription

1 Area, Lattice Points, and Exponential Sums M. N. Huxley College of Cardiff University of Wales CLARENDON PRESS OXFORD 1996

2 Notation Introduction Part I Elementary methods 1. The rational line 1.1 Height 1.2 The Farey sequence 1.3 Lattices and the modular group 1.4 Uniform distribution 1.5 Approximating a given real number 1.6 Uniform Diophantine approximation 2. Polygons and area 2.1 Counting squares 2.2 Jarnflc's polygon 2.3 The discrepancy of a polygon 2.4 Fitting a polygon to a smooth curve 3. The integer points close to a curve 3.1 Introduction 3.2 The reduction step 3.3 The iteration 3.4 Swinnerton-Dyer's method: the convex polygon 3.5 Swinnerton-Dyer's method: counting quadruplets 3.6 Expanding the result 3.7 Points on the curve 4. The rational points close to a curve 4.1 Major and minor sides of the polygon 4.2 Duality 4.3 A linear form often near an integer Part II The Bombieri-Iwaniec method 5. Analytic lemmas 5.1 Bounds for exdonential integrals xi

3 viii 5.2 Partial summation for exponential sums Rounding error sums Poisson summation Evaluating exponential integrals Bilinear and mean square bounds Mean value results The simple exponential sum The lattice point discrepancy The mean square discrepancy The simple exponential sum Motivation Major and minor arcs Poisson summation The large sieve on the minor arcs A preliminary calculation Long major arcs The exponential sum for the lattice point problem Major and minor arcs The major arcs estimate Poisson summation on the minor arcs The large sieve on the minor arcs Exponential sums with a difference Major and minor arcs The major arcs estimate Poisson summation on the minor arcs The large sieve on the minor arcs Exponential sums with modular form coefficients Modular forms The Wilton summation formula Farey arcs Wilton summation on the Farey arcs A large sieve on the Farey arcs Towards a mean square result Jutila's third method 225 Part III The First Spacing Problem: 'integer' vectors 11. The ruled surface method Preparation: divisor functions Families of solutions A comparison argument 249

4 ix 12 The Hardy-Littlewood method Integrals that count The minor arcs The major arcs Extrapolation The First Spacing Problem for the double sum Families of solutions Good and bad families The problem with a perturbing term 283 Part IV The Second Spacing Problem: 'rational' vectors 14. The First and Second Conditions The Coincidence Conditions Magic matrices The Second Condition A family of sums Consecutive minor arcs Parametrizing rational points Coincidence over a short interval Linearizing the Fourth Condition Coincidence over a long interval Extending the Taylor series The Third and Fourth Conditions Counting coincident pairs of minor arcs Sums with congruence conditions Eliminating the centres of the arcs 339 Part V Results and applications 17. Exponential sum theorems The simple exponential sum Exponential sums with a parameter A congruence family of sums Sums with T large Lattice points and area Exponential sums, Integer points and rounding error Lattice points inside a closed curve A family of lattice point problems Rounding error and integration 397

5 x 19. Further results Exponential sums with a difference A major arc estimate Exponential sums with a large second derivative Sums with modular form coefficients Exponential sums Mean value theorems The modular form L-function Applications to the Riemann zeta function Introduction The order of magnitude in the critical strip The mean square Gaps between zeros The twelfth-power moment An application to number theory: prime integer points Preparation Type I double sums Type II double sums Prime numbers in a smooth sequence 463 Part VI Related work and further ideas 23. Related work Integer points close to a curve The Hardy-Littlewood method Other Farey arc arguments Higher dimensions Exponential sums with monomials Further ideas Comments on the method Subdivision without absolute values 479 References 484 Index 491