Explicit methods in number theory

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L-functions play a fundamental role in number theory and encode deep information concerning a wide variety of important arithmetic objects such as prime numbers, elliptic curves, abelian varieties, number fields, and automorphic forms. In spite of their central role, L-functions remain largely mysterious. The goal of this project is to investigate fundamental properties of L-functions, specifically concerning their values and zeros. I will focus my efforts on several areas: the moments and value distribution of the Riemann zeta function and other L-functions, ranks of elliptic curves, identities for L-functions, explicit methods and algorithms for computing the values and zeros of L-functions. The problems that I will address will include: 1) How are the values of the Riemann zeta function distributed? Even the basic question of how large the Riemann zeta function can get, up to given height, is not well understood. Motivated by insights provided by random matrix theory, I believe it should be possible to obtain the full *uniform* asymptotics of the moments and then deduce detailed information regarding the value distribution of the zeta function, including its maximal size. 2) I have discovered a number of useful identities for the Riemann zeta function and Dirichlet L-functions. These formulas are along classical lines but seem to have been missed. They are based on summation methods that I have joyfully explored. The identities can be used to study these L-functions from an analytic point of view, and also for purposes of high precision numerical computation. The formulas also show how different L-functions interrelate. I plan to explore whether any of the methods used can be applied to higher degree L-functions, such as those arising from classical modular forms. 3) I have previously done much work confirming detailed conjectures for the moments of various families of L-functions. An interesting feature occurs for quadratic twists of the zeta function. The theory of multiple Dirichlet series predicts extra lower terms for the cubic and higher moments of these L-functions. Earlier, with my grad student Alderson, we developed algorithms in order to test for these lower terms. Our results were encouraging and do seem to confirm such terms. However, the small constants involved and very noisy remainder term conspire to make it hard to claim conclusive numerical evidence in favour of the extra lower terms. Several avenues of research are proposed. The first is to examine a similar prediction for quadratic twists of elliptic curve L-functions. This will provide a richer data set with which to test for extra lower terms. Another idea, in the case of quadratic Dirichlet L-functions, would be to develop faster algorithms, based on the Fourier expansion of related Eisenstein series, and gather more data. I will also examine function field zeta functions, where a parallel theory suggests the existence of extra lower terms. 4) How large can the rank of an elliptic curve get? Can one find elliptic curves with unusual properties? How efficiently can one compute the rank of an elliptic curve? How often does an elliptic curve have non-trivial rank? These are some of the questions that I plan to investigate. 5) I plan to work on improving algorithms, from the point of view of computational complexity, for computing zeros and values of L-functions. My research will contribute to the field of number theory, providing fundamental and important advances in knowledge, and also resulting in the training of highly qualified personnel.