Several aspects of L-functions

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L--functions play a central role in number theory as many of the deepest questions in the area revolve around them. They encode properties of arithmetic objects, such as the prime numbers. Euclid proved that there are infinitely many prime numbers 2300 years ago, and Gauss conjectured an asymptotic formula for the number of primes 200 years ago. This asymptotic formula was eventually proven by Hadamard and de la Vallée Poussin in 1896 and became known as the Prime Number Theorem. However, obtaining a formula as precise as Gauss conjectured is still an open problem, as it depends on the Riemann Hypothesis (RH), one of the seven Millennium Problems from the Clay Mathematics Institute with a prize of one million dollars. RH is a statement about the zeroes of Riemann zeta function, which is the simplest possible L-function. Many directions arise when considering more general L-functions. The most fundamental questions in this topic are concerned with how large L-functions can be, with the location of their zeroes, and with the values they take at particular numbers (special values). The answers or expected answers to such questions have deep arithmetic significance, such as the Birch and Swinnerton--Dyer conjecture (another Millennium Problem!) and its generalizations. The two far--reaching goals of our research program are concerned with the study of special values of L--functions, and the study of statistics associated to L-functions, particularly of their non-vanishing at certain points. The first direction concerning special values of L--functions has been driven by our focus on Mahler measure of multivariable polynomials. The (logarithmic) Mahler measure of a non-zero polynomial is defined as certain complex integral, and has been found to yield special values of functions of number theoretic significance such as L-functions. One expects that understanding these formulas will yield more information about the nature of the special values. We have been working in the discovery, proof, and understanding of such formulas with the goal of gaining knowledge towards deep conjectures of Beilinson and Bloch on special values of L-functions. We are also studying the dynamical Mahler measure, associated to discrete dynamical systems. The other main direction for our research program revolves around statistics of L-functions. By the work of Montgomery, and then Katz and Sarnak, it is natural to study families of L--functions (sets of L--functions that share a common arithmetic structure) with the expectation that the behavior of the family will provide information about its individual members. We have been focusing on various aspects of the arithmetic statistics of L--functions, in particular for cubic L-functions. The theory is well understood for quadratic L-functions, but much less in known for the cubic case, as it is much more difficult. We have been working on the distribution of values in such families, and in particular non-vanishing results.