On computing Artin functions in the critical strip
Authors:
J. C. Lagarias and A. M. Odlyzko
Journal:
Math. Comp. 33 (1979), 10811095
MSC:
Primary 12A70
MathSciNet review:
528062
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Abstract: This paper gives a method for computing values of certain nonabelian Artin Lfunctions in the complex plane. These Artin Lfunctions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions K of Q. These fields K are the ones for which has an abelian subgroup A of index 2, whose fixed field is complex, and such that there is a for which for all . The key property proved here is that these particular Artin Lfunctions are Hecke (abelian) Lfunctions attached to ring class characters of the imaginary quadratic field and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions. In the special case , where is cubefree, the Artin Lfunction attached to the unique irreducible character of degree 2 of is the quotient of the Dedekind zeta function of the pure cubic field by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For and 12, such representations are used to show that all of the zeroes of these Lfunctions with and are simple and lie on the critical line . These methods currently cannot be used to compute values of Lfunctions with much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.
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DOI:
http://dx.doi.org/10.1090/S00255718197905280629
PII:
S 00255718(1979)05280629
Article copyright:
© Copyright 1979
American Mathematical Society
