Approximation of L-functions associated to Hecke cusp eigenforms

TL;DR

This paper develops a family of approximations for L-functions of Hecke cusp eigenforms, demonstrating their convergence, analyzing error bounds, and successfully locating zeros on the critical line, thus advancing understanding of their analytic properties.

Contribution

It introduces a novel approximation method for L-functions of Hecke cusp eigenforms, extending Matiyasevich's recipe, with proven convergence and applications to zero localization.

Findings

Approximations converge to the true L-function.

Zeros on the critical line are successfully located.

Error bounds are derived using Mellin transforms.

Abstract

We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function and point out the role of an equidistributional notion in ensuring the approximation is well-defined, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives, such as the locations and orders of zeros. Together with the Euler product expansion of the L-function, the family of approximations also encodes some of the key features of the L-function such as its functional equation. As an example, we apply this method to the L-function of the modular discriminant and demonstrate that the approximation successfully locates zeros of the L-function on the critical line. Finally, we derive via Mellin transforms a convolution-type formula which leads to precise error bounds in terms of the incomplete gamma function. This formula can be interpreted as an alternative definition for the approximation and sheds light on Matiyasevich's procedure.