Montgomery H.L. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis

Providence: American Mathematical Society, 1990. — 237 p.This book contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. This book would be an excellent resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics course in number theory.NotationUniform Distribution Free
Qualitative theory
Quantitative relations
Trigonometric approximation
Notesvan der Corput SetsExtremal measures
Relations between α, β[sub(∞)], β[sub()]
Corollaries
A sufficient condition
Intersective sets
Heilbronn sets
NotesExponential Sums I: The Methods of Weyl and van der CorputWeyPs method
van der Corput's method
Exponent pairs
NotesExponential Sums II: Vinogradov's MethodVinogradov's Mean Value Theorem
A bound for Weyl sums
An alternative derivation
NotesAn Introduction to Turan's MethodTuran's First Main Theorem
Fabry's Gap Theorem
Longer ranges of v
Turan's Second Main Theorem
Special coefficients b[sub(n)]
NotesIrregularities of DistributionSquares
Disks
Decay of the Fourier Transform
Families allowing translation, scaling and rotation
NotesMean and Large Values of Dirichlet PolynomialsMean values via trigonometric approximation
Majorant principles
Review of Elementary Operator Theory
Mean values via Hilbert's inequality
Large value estimates
NotesDistribution of Reduced Residue Classes in Short IntervalsA probabilistic model
An approach by Fourier techniques
The fundamental lemma
NotesZeros of L-functionsLeast Character Non-Residues
Clumps of zeros
The Deuring-Heilbronn Phenomenon
NotesSmall Polynomials with Integral CoefficientsThe Gorskov-Wirsing Polynomials
NotesAppendix: Some Unsolved Problems
Uniform Distribution
van der Corput Sets
Weyl Sums
van der Corput's Method
Turan's Method
Irregularities of Distribution
Mean and Large Values of Dirichlet Polynomials
Reduced Residues in Short Intervals
Zeros of L-Functions
Small Polynomials with Integral Coefficients
Character Sums
Diophantine Approximation
Metric Diophantine Approximation
Algebraic Integers
Trigonometric Polynomials
Miscellaneous