In 1990 a conference was held in Manhattan, Kansas, on number theory and harmonic analysis, featuring a series of 10 lectures by H. L. Montgomery. This volume is an expanded version of these lectures. It is not a textbook and it does not aim at completeness; ten exciting subjects are selected, in which active research is going on, there are plenty of open problems and advance can be expected. These subjects are treated rather comprehensively, starting from classical results.
Five of the ten chapters deal with classical topics that have their standard textbooks and monographs. These short surveys do not replace them, but the reader gets an excellent introduction, and even the expert can profit from the newest results and unexpected connections with other fields.
Chapter 1: Uniform distribution. This includes Weyl’s criterion, the Erdős-Turán bound on the discrepancy, Vaaler’s and Selberg’s polynomials.
Chapters 3 and 4: Exponential sums. Weyl’s, van der Corput’s and Vinogradov’s methods for estimating exponential sums are explained. The present state of our knowledge of exponent pairs is related.
Ch. 5: An introduction to Turán’s method. This consists in showing that certain power sums cannot be ‘too small’, complementing the upper estimates treated in the previous chapters.
Ch. 6: Irregularities of distribution. This complements Ch. 1; here lower bounds are sought for various kinds of discrepancy.
Ch. 9: Zeros of \(L\)-functions, is also a classical subject but it is treated from a nonstandard point of view. We do not learn much about zero-free regions and their applications to the distribution of primes; the attention is focused instead on the hypothetical situation “what happens if there is a root somewhere”. One zero of the zeta function near to the line \(\text{Re }z = 1\) induces the existence of others; and also the existence of a real root of an \(L\)-function implies the absence of others from a certain region (Deuring-Heilbronn phenomenon).
The four remaining chapters are devoted to “minor” but exciting subjects.
Ch. 2: Van der Corput sets. These are sets \(H\) with the property that the uniform distribution of \((u_{n + h} - u_ n)\) modulo 1 for all \(h \in H\) implies the u.d. of \(u_ n\). This is connected with the existence of positive trigonometric polynomials with prescribed exponents, the difference-intersective property, and the distribution of \(\alpha h\) (\(h \in H\)) modulo 1 for irrational \(\alpha\).
Ch. 7: Mean and large values of Dirichlet polynomials. This includes an approximate Parseval formula for Dirichlet polynomials on intervals, various estimates for moments, connections with norms of certain operators, Hilbert’s inequality.
Ch. 8: Distribution of reduced residue classes in short intervals. This is a substitute for the study of consecutive primes, where our knowledge is minimal. A probabilistic model is built and some fundamental properties are established with elementary and Fourier methods.
Ch. 10: Small polynomials with integral coefficients. It was observed by Gelfond, rediscovered and generalized by Nair, that a polynomial of degree \(N\) with integral coefficients which is small on \([0,1]\) can yield a bound for \(\pi(N)\). Chebyshev’s bounds can be proved in this way, but, as Gorshkov established, the prime number theorem cannot.
The text is complete with historical remarks, ample references, and a collection of problems from the conference.
The book is a masterpiece of exposition and can be highly recommended to anybody interested in the connections of analysis and number theory.