Previous |  Up |  Next

Article

Keywords:
Dedekind sum; Cochrane sum; Knopp identity
Summary:
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.
References:
[1] Apostol, T. M.: Modular Functions and Dirichlet Series in Number Theory. Springer New York, Heidelberg, Berlin (1976). MR 0422157 | Zbl 0332.10017
[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 0514690
[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 10.1137/0515011 | MR 0728690 | Zbl 0537.10006
[4] Goldberg, L. A.: An elementary proof of the Petersson-Knopp theorem on Dedekind sums. J. Number Theory 12 (1980), 541-542. DOI 10.1016/0022-314X(80)90044-X | MR 0599823 | Zbl 0444.10006
[5] Hall, R. R., Huxley, M. N.: Dedekind sums and continued fractions. Acta Arith. 63 (1993), 79-90. MR 1201620 | Zbl 0785.11027
[6] Knopp, M. I.: Hecke operators and an identity for the Dedekind sums. J. Number Theory 12 (1980), 2-9. DOI 10.1016/0022-314X(80)90067-0 | MR 0566863 | Zbl 0423.10015
[7] Parson, L. A.: Dedekind sums and Hecke operators. Math. Proc. Camb. Philos. Soc. 88 (1980), 11-14. DOI 10.1017/S0305004100057315 | MR 0569629 | Zbl 0435.10005
[8] Pettet, M. R., Sitaramachandrarao, R.: Three-term relations for Hardy sums. J. Number Theory 25 (1987), 328-339. DOI 10.1016/0022-314X(87)90036-9 | MR 0880466 | Zbl 0604.10003
[9] Rademacher, H., Grosswald, E.: Dedekind Sums. The Carus Mathematical Monographs No. 16 The Mathematical Association of America, Washington, D. C. (1972). MR 0357299 | Zbl 0251.10020
[10] Sitaramachandrarao, R.: Dedekind and Hardy sums. Acta Arith. 48 (1987), 325-340. MR 0927374 | Zbl 0635.10002
[11] Zhang, W.: On a Cochrane sum and its hybrid mean value formula. J. Math. Anal. Appl. 267 (2002), 89-96. DOI 10.1006/jmaa.2001.7752 | MR 1886818 | Zbl 1106.11304
[12] Zhang, W.: On a Cochrane sum and its hybrid mean value formula. II. J. Math. Anal. Appl. 276 (2002), 446-457. DOI 10.1016/S0022-247X(02)00501-2 | MR 1944361 | Zbl 1106.11304
[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 10.1016/j.jmaa.2003.09.056 | MR 2020186 | Zbl 1046.11056
[14] Zhang, W., Yi, Y.: On the upper bound estimate of Cochrane sums. Soochow J. Math. 28 (2002), 297-304. MR 1926326 | Zbl 1016.11038
[15] Zheng, Z.: On an identity for Dedekind sums. Acta Math. Sin. 37 (1994), 690-694. Zbl 0842.11017
[16] Zheng, Z.: The Petersson-Knopp identity for homogeneous Dedekind sums. J. Number Theory 57 (1996), 223-230. DOI 10.1006/jnth.1996.0045 | MR 1382748 | Zbl 0847.11021
Partner of
EuDML logo