Geodesic
Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1147-1159
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Let $q$, $h$, $a$, $b$ be integers with $q>0$. The classical and the homogeneous Dedekind sums are defined by $$ s(h,q)=\sum _{j=1}^q\Big (\Big (\frac {j}{q}\Big )\Big )\Big (\Big (\frac {hj}{q}\Big )\Big ),\quad s(a,b,q)=\sum _{j=1}^q\Big (\Big (\frac {aj}{q}\Big )\Big )\Big (\Big (\frac {bj}{q}\Big )\Big ), $$ respectively, where $$ ((x))= \begin {cases} x-[x]-\frac {1}{2}, \text {if $x$ is not an integer};\\ 0, \text {if $x$ is an integer}. \end {cases} $$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$ \gathered \sum _{d\mid n}\sum _{r=1}^d s\Big (\frac {n}{d}a+rq,dq\Big )=\sigma (n)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^d s\Big (\frac {n}{d}a+r_1q,\frac {n}{d}b+r_2q,dq\Big )=n\sigma (n)s(a,b,q), \endgathered $$ where $\sigma (n)=\sum \nolimits _{d\mid n}d$. \endgraf In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.
Let $q$, $h$, $a$, $b$ be integers with $q>0$. The classical and the homogeneous Dedekind sums are defined by $$ s(h,q)=\sum _{j=1}^q\Big (\Big (\frac {j}{q}\Big )\Big )\Big (\Big (\frac {hj}{q}\Big )\Big ),\quad s(a,b,q)=\sum _{j=1}^q\Big (\Big (\frac {aj}{q}\Big )\Big )\Big (\Big (\frac {bj}{q}\Big )\Big ), $$ respectively, where $$ ((x))= \begin {cases} x-[x]-\frac {1}{2}, \text {if $x$ is not an integer};\\ 0, \text {if $x$ is an integer}. \end {cases} $$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$ \gathered \sum _{d\mid n}\sum _{r=1}^d s\Big (\frac {n}{d}a+rq,dq\Big )=\sigma (n)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^d s\Big (\frac {n}{d}a+r_1q,\frac {n}{d}b+r_2q,dq\Big )=n\sigma (n)s(a,b,q), \endgathered $$ where $\sigma (n)=\sum \nolimits _{d\mid n}d$. \endgraf In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.
DOI : 10.1007/s10587-012-0069-7
Classification : 11F20
Keywords: Dedekind sum; Cochrane sum; Knopp identity 
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Liu, Huaning; Gao, Jing. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1147-1159. doi: 10.1007/s10587-012-0069-7

[1] Apostol, T. M.: Modular Functions and Dirichlet Series in Number Theory. Springer New York, Heidelberg, Berlin (1976). | MR | Zbl

[2] Berndt, B. C.: Analytic Eisentein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 303/304 (1978), 332-365. | MR

[3] Berndt, B. C., Goldberg, L. A.: Analytic properties of arithmetic sums arising in the theory of the classical theta-functions. SIAM J. Math. Anal. 15 (1984), 143-150. | DOI | MR | Zbl

[4] Goldberg, L. A.: An elementary proof of the Petersson-Knopp theorem on Dedekind sums. J. Number Theory 12 (1980), 541-542. | DOI | MR | Zbl

[5] Hall, R. R., Huxley, M. N.: Dedekind sums and continued fractions. Acta Arith. 63 (1993), 79-90. | DOI | MR | Zbl

[6] Knopp, M. I.: Hecke operators and an identity for the Dedekind sums. J. Number Theory 12 (1980), 2-9. | DOI | MR | Zbl

[7] Parson, L. A.: Dedekind sums and Hecke operators. Math. Proc. Camb. Philos. Soc. 88 (1980), 11-14. | DOI | MR | Zbl

[8] Pettet, M. R., Sitaramachandrarao, R.: Three-term relations for Hardy sums. J. Number Theory 25 (1987), 328-339. | DOI | MR | Zbl

[9] Rademacher, H., Grosswald, E.: Dedekind Sums. The Carus Mathematical Monographs No. 16 The Mathematical Association of America, Washington, D. C. (1972). | MR | Zbl

[10] Sitaramachandrarao, R.: Dedekind and Hardy sums. Acta Arith. 48 (1987), 325-340. | DOI | MR | Zbl

[11] Zhang, W.: On a Cochrane sum and its hybrid mean value formula. J. Math. Anal. Appl. 267 (2002), 89-96. | DOI | MR | Zbl

[12] Zhang, W.: On a Cochrane sum and its hybrid mean value formula. II. J. Math. Anal. Appl. 276 (2002), 446-457. | DOI | MR | Zbl

[13] Zhang, W., Liu, H.: A note on the Cochrane sum and its hybrid mean value formula. J. Math. Anal. Appl. 288 (2003), 646-659. | DOI | MR | Zbl

[14] Zhang, W., Yi, Y.: On the upper bound estimate of Cochrane sums. Soochow J. Math. 28 (2002), 297-304. | MR | Zbl

[15] Zheng, Z.: On an identity for Dedekind sums. Acta Math. Sin. 37 (1994), 690-694. | Zbl

[16] Zheng, Z.: The Petersson-Knopp identity for homogeneous Dedekind sums. J. Number Theory 57 (1996), 223-230. | DOI | MR | Zbl

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