A problem of Zagier on quadratic polynomials and continued fractions
2016 ◽
Vol 12
(01)
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pp. 121-141
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Keyword(s):
For non-square [Formula: see text] (mod 4), Don Zagier defined a function [Formula: see text] by summing over certain integral quadratic polynomials. He proved that [Formula: see text] is a constant function depending on [Formula: see text]. For rational [Formula: see text], it turns out that this sum has finitely many terms. Here we address the infinitude of the number of quadratic polynomials for non-rational [Formula: see text], and more importantly address some problems posed by Zagier related to characterizing the polynomials which arise in terms of the continued fraction expansion of [Formula: see text]. In addition, we study the indivisibility of the constant functions [Formula: see text] as [Formula: see text] varies.
1992 ◽
Vol 44
(4)
◽
pp. 824-842
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2009 ◽
Vol 29
(1)
◽
pp. 73-109
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2009 ◽
Vol 29
(5)
◽
pp. 1451-1478
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2018 ◽
Vol 2019
(19)
◽
pp. 6136-6161
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1991 ◽
Vol 51
(2)
◽
pp. 324-330
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2001 ◽
Vol 64
(2)
◽
pp. 331-343
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