Abstract
For a totally real algebraic number field k, it is known that every (partial) zeta function of k is a finite sum of Dirichlet series which are regarded as natural generalizations of the Hurwits zeta function (see [1] and [2]). In this note we show that the similar result holds for arbitrary (not necessarily totally real) algebraic number field. At the time of the Bombay Colloquium (1979), H. M. Stark orally communicated to the author that he has obtained such a result for non-real cubic fields. His oral communication was an initial impetus to the present work. The author wishes to express his gratitude to Stark.
Results presented at the time of the Colloquium were relevant to automorphic forms on unitary groups of order 3. However, later the author found several gaps in the proof of those results. Here, another result obtained after the Colloquium is exposed.
Takuro Shintani suddenly passed away on November 14, 1980. Ed.
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References
Shintani, T. On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sec. IA. 23 (1976), 393–417.
Zagier, D. A Kronecker limit formula for real quadratic fields, Math. Ann. 231 (1975), 153–184.
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© 1981 Springer-Verlag Berlin Heidelberg
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Shintani, T. (1981). A Remark on Zeta Functions of Algebraic Number Fields. In: Automorphic Forms, Representation Theory and Arithmetic. Tata Institute of Fundamental Research Studies in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-00734-1_8
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DOI: https://doi.org/10.1007/978-3-662-00734-1_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-10697-5
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