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Zbl 0814.11001
Montgomery, Hugh L.
Ten lectures on the interface between analytic number theory and harmonic analysis. (English)
[B] Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society (AMS). xii, 220 p. \$ 44.00 (1994). ISBN 0-8218-0737-4/pbk

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.\par 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.\par Chapter 1: Uniform distribution. This includes Weyl's criterion, the Erd\H{o}s-Turán bound on the discrepancy, Vaaler's and Selberg's polynomials.\par 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.\par 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.\par Ch. 6: Irregularities of distribution. This complements Ch. 1; here lower bounds are sought for various kinds of discrepancy.\par 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).\par The four remaining chapters are devoted to ``minor'' but exciting subjects.\par Ch. 2: Van der Corput sets. These are sets $H$ with the property that the uniform distribution of $(u\sb{n + h} - u\sb n)$ modulo 1 for all $h \in H$ implies the u.d. of $u\sb n$. 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 $\alpha$.\par 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.\par 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.\par 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.\par The text is complete with historical remarks, ample references, and a collection of problems from the conference.\par The book is a masterpiece of exposition and can be highly recommended to anybody interested in the connections of analysis and number theory.
[I.Z.Ruzsa (Budapest)]

MSC 2000:: *11-02 Research monographs (number theory)
42-02 Research monographs (Fourier analysis)
43-02 Research monographs (abstract harmonic analysis)
11K06 General theory of distribution modulo 1
11K38 Irregularities of distribution
11K70 Harmonic analysis and almost periodicity
11L03 Trigonometric and exponential sums, general
11L07 Estimates on exponential sums
11L15 Weyl sums
11M20 Real zeros of L(s,chi)
11M26 Nonreal zeros of zeta(s) and L(s,chi)
11N05 Distribution of primes
11N25 Distribution of integers with specified multiplicative constraints
11N30 Turan theory
11N69 Distribution of integers in special residue classes
42A05 Trigonometric polynomials
42A10 Trigonometric approximation
11N32 Primes represented by polynomials

Keywords: analytic number theory; uniform distribution; exponential sums; irregularities of distribution; zeros of $L$-functions; distribution of reduced residue classes in short intervals; small polynomials with integral coefficients; harmonic analysis; surveys; Turán's method; Van der Corput sets; values of Dirichlet polynomials

Cited in: Zbl 1057.11038 Zbl 0997.11065 Zbl 1015.11064 Zbl 0970.11043 Zbl 0949.11039

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