Analytic Theory of L-functions and Modular Forms

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The main part in this proposal concerns the relationship between mean-values of L-functions, the distribution of zeros of L-functions, and the distribution of the prime numbers. The investigator and his colleagues study mean-values of L-functions times a "mollifier," in order to prove an equivalence between the mean-value and the distribution of zeros of the L-function predicted by random matrix theory. The investigator and his colleagues also study the connection between the distribution of zeros and variations in the distribution of primes in short intervals. In a separate project the investigator develops computer programs to construct Maass forms on Hecke congruence subgroups. The resulting data confirms the predictions of random matrix theory.The work in this proposal is motivated by recently discovered connections between number theory and physics. These connections reveal a relationship between classical number theory (mathematics) and quantum chaos (physics). The main tool used to exhibit the connection is Random Matrix Theory, a subject which arose as an explanation for certain experimental results in nuclear physics. Random Matrix Theory has since been found to be related to other physical phenomena, as well as several areas of purely theoretical mathematics. By exploiting this connection with Random Matrix Theory it is possible to use mathematics to explain phenomena observed in physical experiments, and to use the underlying physics as motivation for discovering new mathematics. In this way, more progress is possible by combining the two approaches than could be achieved separately. The majority of projects in this grant will be carried out with the assistance of undergraduate research students.