Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Liu, Huaning ; Gao, Jing

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1147-1159.

Résumé

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 $\sigma (n)={\sum }_{d\mid n}d$. \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.

Détail

DOI : 10.1007/s10587-012-0069-7

Classification : 11F20
Mots-clés : Dedekind sum; Cochrane sum; Knopp identity