ON A PROBLEM OF FINDING NON-TRIVIAL ZEROS OF DIRICHLET L-FUNCTIONS IN NUMBER FIELDS

Abstract

There is a numeric algorithm for finding non-trivial zeros of regular Dirichlet L-functions. This algorithm is based on a construction of Dirichlet polynomials which approximate these L-functions in any rectangle in the critical strip with exponential speed. This result does not hold for Dirichlet L-functions in number fields, because if it did, a power series with the same coefficients as the Dirichlet series defining the L-function would converge to a function which is holomorphic at 1, however, it is known that that such power series in case of a number field different from the field of rational numbers can’t be continued analytically past its convergence boundary. Consequently, we need to develop a new numerical algorithm for finding non-trivial zeros of Dirichlet L-functions in number fields. This problem is discussed in this paper. We show that there exists a sequence of Dirichlet polynomials which approximate a Dirichlet L-function in a number field faster than any power function in any rectangle inside the critical strip. We also provide an explicit construction of approximating Dirichlet polynomials, whose zeros coincide with those of a Dirichlet L-function in the specified rectangle, for an L-function, if it can be split into a product of classical L-functions. Additionally we discuss some questions related to the construction of such polynomials for arbitrary Dirichlet L-functions.