Abstract
In this paper, we present a new reciprocity formula for certain Hardy sums and study a computational problem of one type mean value containing certain Hardy sums and the two-term exponential sum with the help of the properties of Gauss sums and Dirichlet \(L-\)function.
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References
B.C. Berndt, Analytic Eisenstein series, theta functions and series relations in the spirit of Ramanujan, J. Reine Angew. Math. 303/304, 332–365, (1978).
T. Cochrane, Exponential sums modulo prime powers, Acta Arith. 101, 131–149, (2002).
T. Cochrane and C. Pinner, A further refinement of Mordell’s bound on exponential sums, Acta Arith. 116, 35–41, (2005).
T. Cochrane and C. Pinner, Using Stepanov’s method for exponential sums involving rational functions, J. Number Theory 116, 270–292, (2006).
J.B. Conrey, E. Fransen, R. Klein, and C. Scott, Mean values of Dedekind sums, J. Number Theory 56, 214–226, (1996).
M.C. Dağlı, 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. Dağlı, On some identities involving certain Hardy sums and a Kloosterman sum, Ukr. Math. J. 72(11), 1724–1732 (2021).
X. Du and L. Zhang, On the Dedekind sums and its new reciprocity formula, Miskolc Math. Notes 19 235–239, (2018).
L.A. Goldberg, Transformations of theta-functions and analogues of Dedekind sums, Ph.D. thesis, University of Illinois, Urbana 1981.
D. Han and T. Wang, On the Dedekind sums and two-term exponential sums, Chin. Ann. Math. Ser. B 36, 503–612, (2015).
D. Han and W. Wang, Some new identities involving Dedekind sums and the Ramanujan sum, Ramanujan J. 35, 253–262 (2014).
C. Jia, On the mean value of Dedekind sums, J. Number Theory 87, 173–188, (2001).
L.A. Parson, Dedekind sums and Hecke Operators, Math. Proc. Cambridge Phil. Soc. 88, 11–14, (1980).
W. Peng and T. Zhang, Some identities involving certain Hardy sum and Kloosterman sum, J. Number Theory 165, 355–362, (2016).
H. Rademacher and E. Grosswald, Dedekind sums, Math. Assoc. of America, Washington, D.C. 1972.
R. Sitaramachandrarao, Dedekind and Hardy sums, Acta Arith. XLIII, 325–340, (1987).
H. Walum, An exact formula for an average of L-series, Ill. J. Math. 26, 1–3, (1982).
T. Wang and X. Pan, The mean value involving Dedekind sums and two-term exponential sums, Sci. China Math. 55, 557–565, (2012).
T. Wang and W. Zhang, On the hybrid mean value of Dedekind sums and two-term exponential sums, Front. Math. China 6 (3), 557–563, (2011).
H. Zhang and W. Zhang, On the identity involving certain Hardy sums and Kloosterman sums, J. Inequal. Appl. 52, 2014, 9 pp.
J. Zhang and W. Zhang, On the general Dedekind sums and two-term exponential sums, Sci. World J., Vol. 2014, Article ID 146926, 2014, 4 pages.
W. Zhang, On the mean values of Dedekind sums, J. Theor. Nombres Bordeaux 8, 429–442, (1996).
W. Zhang, A note on the mean square value of the Dedekind sums, Acta Math. Hung. 86, 275–289, (2000).
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Dağlı, M.C. New identities involving certain Hardy sums and two-term exponential sums. Indian J Pure Appl Math 54, 841–847 (2023). https://doi.org/10.1007/s13226-022-00302-0
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DOI: https://doi.org/10.1007/s13226-022-00302-0