Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2020; 57(4): 845-871

Published online July 1, 2020 https://doi.org/10.4134/JKMS.j190406

Copyright © The Korean Mathematical Society.

Equidistribution of higher dimensional generalized Dedekind sums and exponential sums

Hi-joon Chae, Byungheup Jun, Jungyun Lee

Hongik University; Ulsan National Institute of Science and Technology; Kangwon National University

Abstract

We consider generalized Dedekind sums in dimension $n$, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in $\FR/\FZ$.

Keywords: generalized Dedekind sums, Todd series, exponential sums, equidistribution

MSC numbers: 11F20, 11L03, 14M25

Supported by: The first author was supported by 2019 Hongik University Research Fund.
The second author was supported by NRF-2018R1D1A1A02085748.
The third author was supported by 2019 Research Grant from Kangwon National University and NRF-2017R1A6A3A11030486.