On a problem of finding non-trivial zeros of Dirichlet $L$-functions in number fields

V. N. Kuznetsov ; V. A. Matveev

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 144-154

Résumé

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. Bibliography: 11 titles.

Détail

Keywords: Dirichlet characters, Dirichlet $L$-functions in number fields, non-trivial zeros of $L$-functions.