zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Ten lectures on the interface between analytic number theory and harmonic analysis. (English) Zbl 0814.11001
Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society (AMS). xii, 220 p. $ 44.00 (1994).
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 $(u\sb{n + h} - u\sb n)$ 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 $\alpha$. 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.

11-02Research monographs (number theory)
42-02Research monographs (Fourier analysis)
43-02Research monographs (abstract harmonic analysis)
11K06General theory of distribution modulo 1
11K38Irregularities of distribution
11K70Harmonic analysis and almost periodicity
11L03Trigonometric and exponential sums, general
11L07Estimates on exponential sums
11L15Weyl sums
11M20Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
11M26Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
11N05Distribution of primes
11N25Distribution of integers with specified multiplicative constraints
11N30Turán theory
11N69Distribution of integers in special residue classes
42A05Trigonometric polynomials, inequalities, extremal problems
42A10Trigonometric approximation
11N32Primes represented by polynomials