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Special values of partial zeta functions of real quadratic fields at nonpositive integers and the Euler-Maclaurin formula


Authors: Byungheup Jun and Jungyun Lee
Journal: Trans. Amer. Math. Soc. 368 (2016), 7935-7964
MSC (2010): Primary 11E41, 11M41, 11R11, 11R29, 11R80
DOI: https://doi.org/10.1090/tran/6679
Published electronically: February 12, 2016
MathSciNet review: 3546789
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Abstract: We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field in terms of the positive continued fraction of the reduced element defining the ideal. We apply the integral expression of the partial zeta value due to Garoufalidis-Pommersheim (2001) using the Euler-Maclaurin summation formula for a lattice cone associated to the ideal. From the additive property of Todd series w.r.t. the (virtual) cone decomposition arising from the positive continued fraction of the reduced element of the ideal, we obtain a polynomial expression of the partial zeta values with variables given by the coefficient of the continued fraction. We compute the partial zeta values explicitly for $ s=0,-1,-2$ and compare the result with earlier works of Zagier (1977) and Garoufalidis-Pommersheim (2001). Finally, we present a way to construct Yokoi-Byeon-Kim type class number one criterion for some families of real quadratic fields.


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Additional Information

Byungheup Jun
Affiliation: Department of Mathematics, Yonsei University, Yonsei-ro 50, Seodaemun-gu, Seoul 120-749, Korea
Address at time of publication: Department of Mathematical Sciences, UNIST, UNIST-gil 50, Ulsan 689-798, Korea
Email: bhjun@unist.ac.kr

Jungyun Lee
Affiliation: Department of Mathematics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 120-750, Korea
Email: lee9311@ewha.ac.kr

DOI: https://doi.org/10.1090/tran/6679
Received by editor(s): October 29, 2013
Received by editor(s) in revised form: February 2, 2015
Published electronically: February 12, 2016
Additional Notes: The work of the first author was supported by NRF grant (NRF-2015R1D1A1A09059083) and Samsung Science Technology Foundation grant BA1301-03
The work of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (2011-0023688) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827)
Article copyright: © Copyright 2016 American Mathematical Society