On the mean value of the generalized Dedekind sum and certain generalized Hardy sums weighted by the Kloosterman sum
Abstract
UDC 511
We study a hybrid mean-value problem related to the generalized Dedekind sum, certain generalized Hardy sums, and Kloosterman sum and obtain several meaningful conclusions by means of the analytic method and the properties of the character sum and the Gauss sum.
References
T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York (1976). DOI: https://doi.org/10.1007/978-1-4757-5579-4
B. C. Berndt, Analytic Eisenstein series, theta functions and series relations in the spirit of Ramanujan}, J. reine und angew. Math., 303/304, 332–365 (1978). DOI: https://doi.org/10.1515/crll.1978.303-304.332
M. Can, Some arithmetic on the Hardy sums $s_{2}(h,k)$ and $s_{3}(h,k)$, Acta Math. Sin. (Engl. Ser.), 20, № 2, 193–200 (2004). DOI: https://doi.org/10.1007/s10114-004-0324-4
J. B. Conrey, E. Fransen, R. Klein, C. Scott, Mean values of Dedekind sums, J. Number Theory, 56, 214–226 (1996). DOI: https://doi.org/10.1006/jnth.1996.0014
M. C. Daǧlı, On the hybrid mean-value of generalized Dedekind sums, generalized Hardy sums and Ramanujan sum, Bull. Math. Soc. Sci. Math. Roumanie, 63, № 4, 325–333 (2020).
M. C. Daǧlı, On some identities involving certain Hardy sums and a Kloosterman sum, Ukr. Math. J., 72, № 11, 1724–1732 (2021). DOI: https://doi.org/10.1007/s11253-021-01883-5
T. Estermann, On Kloostermann's sum, Mathematica, 8, 83–86 (1961). DOI: https://doi.org/10.1112/S0025579300002187
L. A. Goldberg, Transformations of theta-functions and analogues of Dedekind sums, Ph. D. Thesis, Univ. Illinois, Urbana (1981).
L. Huan, J. Wang, T. Wang, An identity involving Dedekind sums and generalized Kloosterman sums, Czech. Math. J., 62, № 4, 991–1001 (2012). DOI: https://doi.org/10.1007/s10587-012-0078-6
S. Hu, D. Kim, M.-S. Kim, On reciprocity formula of Apostol–Dedekind sum with quasi-periodic Euler functions, J. Number Theory, 162, 54–67 (2016). DOI: https://doi.org/10.1016/j.jnt.2015.10.022
S. Hu, M.-S. Kim, The $p$-adic analytic Dedekind sums, J. Number Theory, 171, 112–127 (2017). DOI: https://doi.org/10.1016/j.jnt.2016.07.022
M.-S. Kim, J.-W. Son, On generalized Dedekind sums involving quasi-periodic Euler functions, J. Number Theory, 144, 267–280 (2014). DOI: https://doi.org/10.1016/j.jnt.2014.05.022
H. Liu, W. Zhang, Generalized Dedekind sums and generalized Hardy sums, Acta Math. Sinica (Chin. Ser.), 49, 999–1008 (2006).
Y. N. Liu, W. Zhang, A hybrid mean-value related to the Dedekind sums and Kloosterman sums, Acta Math. Sin. (Engl. Ser.), 27, № 3, 435–440 (2011). DOI: https://doi.org/10.1007/s10114-010-9192-2
W. Peng, T. Zhang, Some identities involving certain Hardy sum and Kloosterman sum, J. Number Theory, 165, 355–362 (2016). DOI: https://doi.org/10.1016/j.jnt.2016.01.028
Y. Simsek, Relations between theta functions, Hardy sums, Eisenstein and Lambert series in the transformation formulae of $logη_{g,h}(z)$, J. Number Theory, 99, 338–360 (2003). DOI: https://doi.org/10.1016/S0022-314X(02)00072-0
Q. Tian, On the hybrid mean-value of generalized Dedekind sums, generalized Hardy sums and Kloosterman sums; http://arxiv.org/abs/1809.07538.
H. Walum, An exact formula for an average of $L$-series, Illinois J. Math., 26, 1–3 (1982). DOI: https://doi.org/10.1215/ijm/1256046895
Z. Wenpeng, L. Yanni, A hybrid mean-value related to the Dedekind sums and Kloosterman sums, Sci. China Math., 53, 2543–2550 (2010). DOI: https://doi.org/10.1007/s11425-010-3153-1
H. Zhang, W. Zhang, On the identity involving certain Hardy sums and Kloosterman sums, J. Inequal. and Appl., 52 (2014). DOI: https://doi.org/10.1186/1029-242X-2014-52
W. Zhang, On the mean-values of Dedekind sums, J. Théor. Nombres Bordeaux, 8, 429–442 (1996). DOI: https://doi.org/10.5802/jtnb.179
Copyright (c) 2023 Muhammet Cihat Dağlı
This work is licensed under a Creative Commons Attribution 4.0 International License.
