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Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the subject and to present recent work on the many active areas of the subject (the discussion on previous blog posts here have mostly focused on advances in the study of the distribution of the prime numbers, but there have been many other notable recent developments too, such as refinements of the circle method, a deeper understanding of the asymptotics of bounded multiplicative functions and of the “pretentious” approach to analytic number theory, more “analysis-friendly” formulations of the theorems of Deligne and others involving trace functions over fields, and new subconvexity theorems for automorphic forms, to name a few). Like any other semester MSRI program, there will be a number of workshops, seminars, and similar activities taking place while the members are in residence. I’m personally looking forward to the program, which should be occurring in the midst of a particularly productive time for the subject. Needless to say, I (and the rest of the organising committee) plan to be present for most of the program.

Applications for Postdoctoral Fellowships, Research Memberships, and Research Professorships for this program (and for other MSRI programs in this time period, namely the companion program in Harmonic Analysis and the Fall program in Geometric Group Theory, as well as the complementary program in all other areas of mathematics) have just opened up today. Applications are open to everyone (until they close on Dec 1), but require supporting documentation, such as a CV, statement of purpose, and letters of recommendation from other mathematicians; see the application page for more details.

[*This post is authored by Emmanuel Kowalski*.]

This post may be seen as complementary to the post “The parity problem in sieve theory“. In addition to a survey of another important sieve technique, it might be interesting as a discussion of some of the foundational issues which were discussed in the comments to that post.

Many readers will certainly have heard already of one form or another of the “large sieve inequality”. The name itself is misleading however, and what is meant by this may be something having very little, if anything, to do with sieves. What I will discuss are genuine sieve situations.

The framework I will describe is explained in the preprint arXiv:math.NT/0610021, and in a forthcoming Cambridge Tract. I started looking at this first to have a common setting for the usual large sieve and a “sieve for Frobenius” I had devised earlier to study some arithmetic properties of families of zeta functions over finite fields. Another version of such a sieve was described by Zywina (“The large sieve and Galois representations”, preprint), and his approach was quite helpful in suggesting more general settings than I had considered at first. The latest generalizations more or less took life naturally when looking at new applications, such as discrete groups.

Unfortunately (maybe), there will be quite a bit of notation involved; hopefully, the illustrations related to the classical case of sieving integers to obtain the primes (or other subsets of integers with special multiplicative features) will clarify the general case, and the “new” examples will motivate readers to find yet more interesting applications of sieves.

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