Skip to main content Accessibility help
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

78 results

Page 1 of 4

  • 19 - The Large Sieve

  • Hugh L. Montgomery, University of Michigan, Ann Arbor, Robert C. Vaughan, Pennsylvania State University
  • Book:
    Multiplicative Number Theory II
    Published online:
    02 April 2026
    Print publication:
    09 April 2026, pp 149-188

  • Summary

    The large sieve takes various forms, as a mean square upper bound for a trigonometric polynomial at well-spaced points, as a mean square upper bound for the distribution of a set of integers into arithmetic progressions, and as a mean square upper bound for character sums. We take the trigonometric form to be fundamental and derive the other versions from it.

  • 20 - Primes in Arithmetic Progressions: III

  • Hugh L. Montgomery, University of Michigan, Ann Arbor, Robert C. Vaughan, Pennsylvania State University
  • Book:
    Multiplicative Number Theory II
    Published online:
    02 April 2026
    Print publication:
    09 April 2026, pp 189-225

  • Summary

    Our best unconditional bounds for ψ(x, χ), such as Theorem 11.16, are not very good owing to our rather limited knowledge of the zero-free region of L(s, χ). It is possible to extend the range for the modulus q of χ at the expense of the error term or by including zeros close to 1. See Corollary 11.20 or the chapter on Vinogradov’s mean value theorem in the forthcoming Volume III. If we assume GRH, then we have a much better estimate (cf. Theorem 13.7). However, in some situations, a good bound for an average of |ψ (x, χ) | is all that is required, and such bounds can be obtained unconditionally by combining our methods of Chapter 17 with the large sieve.

  • 21 - Sieves of Fixed Dimension and Applications

  • Hugh L. Montgomery, University of Michigan, Ann Arbor, Robert C. Vaughan, Pennsylvania State University
  • Book:
    Multiplicative Number Theory II
    Published online:
    02 April 2026
    Print publication:
    09 April 2026, pp 226-304

  • Summary

    In this chapter we return to the topic of Chapter 3, (small) sieves, which we now treat, at least initially, in some generality. However our objective is to give nothing more than an introduction and some applications to what has become a vast and complex subject. Readers who wish to see the many aspects of the subject in more detail are advised to consult the standard reference on the subject, Friedlander & Iwaniec (2010).

  • 22 - Bounded Gaps between Primes

  • Hugh L. Montgomery, University of Michigan, Ann Arbor, Robert C. Vaughan, Pennsylvania State University
  • Book:
    Multiplicative Number Theory II
    Published online:
    02 April 2026
    Print publication:
    09 April 2026, pp 305-334

  • Summary

    An important role is played in the most recent developments on gaps between primes by suitable sets of prime k-tuples. Thus before proceeding with this chapter the reader would be well advised to review the contents of §18.5. The principal idea is to use artefacts from sieve theory, especially the Selberg sieve, not directly in the form of a sieve but as a means to increase the likelihood that certain constellations of 𝑘-tuples have relatively few prime factors.

Multiplicative Number Theory II
  • Primes and Sieves
  • Hugh L. Montgomery, Robert C. Vaughan
  • Published online:
    02 April 2026
    Print publication:
    09 April 2026
  • This long-anticipated work shares the aims of its celebrated companion: namely, to provide an introduction for students and a reference for researchers to the techniques, results, and terminology of multiplicative number theory. This volume builds on the earlier one (which served as an introduction to basic, classical results) and focuses on sieve methods. This area has witnessed a number of major advances in recent years, e.g. gaps between primes, large values of Dirichlet polynomials and zero density estimates, all of which feature here. Despite the fact that the book can serve as an entry to contemporary mathematics, it remains largely self-contained, with appendices containing background or material more advanced than undergraduate mathematics. Again, exercises, of which there is a profusion, illustrate the theory or indicate ways in which it can be developed. Each chapter ends with a thorough set of references, which will be essential for all analytic number theorists.

Page 1 of 4