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This article is cited in 2 scientific papers (total in 2 papers)
On a problem of finding non-trivial zeros of Dirichlet $L$-functions in number fields
V. N. Kuznetsov, V. A. Matveev Saratov State University named after N. G. Chernyshevsky
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.
Bibliography: 11 titles.
Keywords:
Dirichlet characters, Dirichlet $L$-functions in number fields, non-trivial zeros of $L$-functions.
Received: 13.05.2015
Citation:
V. N. Kuznetsov, V. A. Matveev, “On a problem of finding non-trivial zeros of Dirichlet $L$-functions in number fields”, Chebyshevskii Sb., 16:2 (2015), 144–154
Linking options:
https://www.mathnet.ru/eng/cheb395 https://www.mathnet.ru/eng/cheb/v16/i2/p144
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Abstract page: | 254 | Full-text PDF : | 93 | References: | 56 |
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