Zeta-Functions, L-Functions, and Random Matrix Theory

DMS-0201457 Steve GonekAbstractThe investigator and his colleagues are studying problems in analyticnumber theory centered mainly on the Riemann-zeta function and L-functions,and relations between their zeros and the primes. Two projects focus on theremarkable, recently discovered applications of random matrix theory tothe zeta-function. Keating and Snaith's characteristic polynomial model ofthe zeta-function is providing heuristic answers to many previouslyintractable problems in analytic number theory. However, a serious drawbackof the model is that it does not contain the primes. Instead, they have tobe inserted in an ad hoc manner with each new application. The investigatorand his colleagues have a new model that explicitly integrates the primesand zeros in a most natural way. They are applying it to a variety ofproblems, such as moment and order estimates for the zeta-function, andthey expect it to give them new insights into the connections between thezeros and the primes. A related project explores the relations between theGaussian Unitary Ensemble Conjecture on the zeros, the distribution ofprimes and almost-primes, and mean-value theorems involving the Riemannzeta-function. They also study geometrical aspects of the zeta-function,such as its curvature near the critical line, and the size of gaps betweenzeros. Two final problems lie in an altogether different area of numbertheory. These concern additive patterns of elements in the multiplicativegroup of a finite field and the related question of the value distributionof incomplete exponential sums containing multiplicative characters.This is a project in the area of mathematics known as number theory. The fundamental questions of interest in number theory have to do with the structure of numbers, and in particular the prime numbers, as these are the fundamental building blocks of arithmetic and, therefore, of much of mathematics. Many of the most important questions in this area are so intractable that it is impossible even to guess correct answers to them. Recently, a remarkable partnership has developed between theoretical physicists and analytic number theorists, which is succeeding in answering some of these questions. At the center of this collaboration is a model of something called the Riemann zeta-function, which is a special mathematical function known for over a century to encode within its properties a great deal of information about prime numbers. This model is based on the theory of random matrices, objects previously used to model complicated physical systems such as heavy nuclei. Although the model has been quite successful, it has the serious drawback of not containing the prime numbers which, after all, are the principal objects of interest. The investigator and his colleagues have now developed a model that does integrate the primes and zeros in a most natural way, and a main goal of this project is to explore further applications of our new model. A related project concerns arithmetic and analytic consequences of a widely believed conjecture about the zeros of the zeta-function. Using recent developments in the field, we also investigate geometrical aspects of the zeta-function such as its curvature, and gap sizes between its zeros. Two problems in a different area of number theory concern the structure of finite fields, objects with important applications to coding theory and cryptography.

DMS-0201457 Steve GonekAbstractThe investigator and his colleagues are studying problems in analyticnumber theory centered mainly on the Riemann-zeta function and L-functions,and relations between their zeros and the primes. Two projects focus on theremarkable, recently discovered applications of random matrix theory tothe zeta-function. Keating and Snaith's characteristic polynomial model ofthe zeta-function is providing heuristic answers to many previouslyintractable problems in analytic number theory. However, a serious drawbackof the model is that it does not contain the primes. Instead, they have tobe inserted in an ad hoc manner with each new application. The investigatorand his colleagues have a new model that explicitly integrates the primesand zeros in a most natural way. They are applying it to a variety ofproblems, such as moment and order estimates for the zeta-function, andthey expect it to give them new insights into the connections between thezeros and the primes. A related project explores the relations between theGaussian Unitary Ensemble Conjecture on the zeros, the distribution ofprimes and almost-primes, and mean-value theorems involving the Riemannzeta-function. They also study geometrical aspects of the zeta-function,such as its curvature near the critical line, and the size of gaps betweenzeros. Two final problems lie in an altogether different area of numbertheory. These concern additive patterns of elements in the multiplicativegroup of a finite field and the related question of the value distributionof incomplete exponential sums containing multiplicative characters.This is a project in the area of mathematics known as number theory. The fundamental questions of interest in number theory have to do with the structure of numbers, and in particular the prime numbers, as these are the fundamental building blocks of arithmetic and, therefore, of much of mathematics. Many of the most important questions in this area are so intractable that it is impossible even to guess correct answers to them. Recently, a remarkable partnership has developed between theoretical physicists and analytic number theorists, which is succeeding in answering some of these questions. At the center of this collaboration is a model of something called the Riemann zeta-function, which is a special mathematical function known for over a century to encode within its properties a great deal of information about prime numbers. This model is based on the theory of random matrices, objects previously used to model complicated physical systems such as heavy nuclei. Although the model has been quite successful, it has the serious drawback of not containing the prime numbers which, after all, are the principal objects of interest. The investigator and his colleagues have now developed a model that does integrate the primes and zeros in a most natural way, and a main goal of this project is to explore further applications of our new model. A related project concerns arithmetic and analytic consequences of a widely believed conjecture about the zeros of the zeta-function. Using recent developments in the field, we also investigate geometrical aspects of the zeta-function such as its curvature, and gap sizes between its zeros. Two problems in a different area of number theory concern the structure of finite fields, objects with important applications to coding theory and cryptography.