On Generalized Hardy Sums s5(h, k)
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The aim of this paper is to study generalized Hardy sums s5(h, k). By using mediants and the adjacent difference of Farey fractions, we establish a relationship between s5(h, k) and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums s5,p(h,k). A relationship between s5,p(h, k) and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between s5(h,k) and Lambert series.
KeywordsZeta Function Hurwitz Zeta Function Bernoulli Function Farey Fraction Lambert Series
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- 2.R. Sitaramachandraro, “Dedekind and Hardy sums,” Acta Arithm., 48 (1978).Google Scholar
- 3.G. H. Hardy, “On certain series of discontinuous functions, connected with the modular functions,” Quart. J. Math. 36, 93–123 (1905).Google Scholar
- 4.B. C. Berndt, “Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan,” J. Reine Angew. Math., 303/304, 332–150 (1978).Google Scholar
- 5.L. A. Goldberg, Transformation of Theta-Functions and Analogues of Dedekind Sums, Thesis, Univ. Illinois Urbana (1981).Google Scholar
- 6.E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge (1962).Google Scholar
- 8.T. M. Apostol and T. H. Vu, “Elementary proofs of Berndt’s reciprocity laws,” Pacif. J. Math., 98, 17–23 (1982).Google Scholar
- 9.Y. Simsek, Dedekind ve Hardy Toplamlarinin Genellestrilmesi, Thesis, Cukurova Univ., Adana (1993).Google Scholar
- 10.Y. Simsek, “Theorems on three-term relations for Hardy sums,” Turkish J. Math., 22, 153–162 (1998).Google Scholar
- 12.Y. Simsek, “A note on Dedekind sums,” Bull. Cal. Math. Soc., 85, 567–572 (1993).Google Scholar