The Higher Rank Selberg Sieve and Applications

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In 1947, Atle Selberg discovered a new method in sieve theory which revolutionized the subject.  This method is now called the Selberg sieve and has been used in a spectrum of applications ranging from the classical twin prime problem to more sophisticated questions of counting points on algebraic varieties.  Recently, a special case of a "higher rank'' version of the Selberg sieve was applied by Maynard and Tao (independently) to improve and simplify upon Zhang's ground breaking work regarding infinitely many bounded gaps between consecutive prime numbers.  In joint work with my doctoral student, Akshaa Vatwani, I have developed a general higher rank version of the classical Selberg sieve.  The work of Maynard and Tao now appears as a special case of this more general sieve.  Clearly, there will be further applications of this new sieve method and we plan to apply it to an assortment of problems in the coming years.  It looks as if there will be some potential applications to the Artin primitive root problem.  In addition, one can formulate also a number field version of this higher rank sieve.  My student is already looking at this possibility in her doctoral thesis currently in progress, and has obtained some interesting results regarding bounded gaps between Gaussian primes.  That is, there is a fixed number B such that there are infinitely many Gaussian primes a+bi and c+di such that |a-c| and |b-d| are both bounded by B.  We also have a new proof of the Tauberian theorem and we expect more results in this setting.***Clearly, these recent discoveries represent a cusp in sieve theory.  Indeed, the relationship between the higher rank Selberg sieve and the classical Selberg sieve is similar to the discovery of muti-variable calculus and one-variable calculus.  For example, in the classical Selberg sieve, in an attempt to prove that there are infinitely many twin primes, the sieve method was applied to the single sequence n(n+2) instead of the two-tuple sequence (n, n+2).  It is historically interesting that Selberg had suggested the "higher rank'' approach to the sieve as far back as 1969, at the end of one of his papers, but clearly the idea went unnoticed until it was recently resurrected by Maynard and Tao.  Though one can study functions of several variables using a one-variable theory, there is a richer structure in the multi-variable theory.  Similar is the case with the classical sieve and the higher rank sieve. ***We also expect other applications of the higher rank Selberg sieve to the study of gaps between primes satisfying Chebotarev conditions especially when the base field is not the rational number field.  My doctoral student, Peng-Jie Wong, is now investigating Artin L-series and we are studying improvements to the Chebotarev density theorem, with a view to applying number field versions of the higher rank sieve.  Related questions, but in the setting of elliptic curves, are being studied by my third doctoral student, Francois Seguin.  **