L-functions and Operators

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Dr. Lagarias proposes to study various analytic problems related to zeta and L-functions in number theory. There are three topics. The first topic is the study of Li coefficients as a family and their compatiblity with automorphic representations. The second topic concerns interpreting zeros of automorphic L-functions in the framework of Hilbert spaces of entire functions, and study of associated operators. The third topic is the study of generalizations of a two-variable zeta function of van der Geer and Schoof involving automorphic forms. This proposal is in the general area of mathematics called number theory. This work studies various problems associated to the distribution of prime numbers. These include possible description of some of its properties in terms of equations in mathematical physics. Number theory is an old field which in recent years has provided many useful applications in communications, coding theory and cryptography. The performance of some cryptographic procedures-necessary to secure internet communications - are related to subtle problems on the distribution of prime numbers. The questions treated here overlap different fields of mathematics and this research may stimulate interactions between researchers in these fields.