Truncated Product Representations for L-Functions in the Hyperelliptic Ensemble

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordWe investigate the approximation of quadratic Dirichlet L-functions over function fields by truncations of their Euler products. We first establish representations for such L-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an L-function away from its zeros, and that, when the length of the product tends to infinity, we recover the original L-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet L-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet L-function over a function field, an auxiliary function based on the approximate functional equation that equals the L-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions, prove the Riemann hypothesis holds for them, and that their zeros are related to those of the associated L-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated L-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.JCA was partially supported by a Research in Pairs - Scheme 4 London Mathematical Society grant. SMG was supported in part by National Science Foundation Grant DMS-1200582. JPK gratefully acknowledges support under EPSRC Programme Grant EP/K034383/1 (LMF: L-Functions and Modular Forms) and a Royal Society Wolfson Research Merit Award

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Truncated product representations for L-functions in the hyperelliptic ensemble

We investigate the approximation of quadratic Dirichlet L -functions over function fields by truncations of their Euler products. We first establish representations for such L -functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an L -function away from its zeros and that, when the length of the product tends to infinity, we recover the original L -function. We also obtain explicit expressions for the arguments of quadratic Dirichlet L -functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet L -function over a function field, an auxiliary function based on the approximate functional equation that equals the L -function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated L -function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated L -function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.</p