Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/some-generalizations-of-knopp-s-identity-mfupw8Z0xI
References (9)
-
Zhiyong Zheng (1996)
The Petersson–Knopp Identity for the Homogeneous Dedekind SumsJournal of Number Theory, 57
-
Beifang Chen, Zhi-Wei Sun (2002)
Generalizations of Knopp's IdentityJournal of Number Theory, 97
-
Larry Goldberg (1980)
An elementary proof of the Petersson-Knopp theorem on Dedekind sumsJournal of Number Theory, 12
-
T. Apostol (1976)
Modular Functions and Dirichlet Series in Number Theory -
(1972)
On a theorem of Dedekind sums -
K. Walker (1992)
A. Dedekind Sums -
M. Knopp (1980)
Hecke operators and an identity for the dedekind sumsJournal of Number Theory, 12
-
(1996)
Petersson-Knopp identity for the Homogeneous Dedekind -
L. Parson (1980)
Dedekind sums and Hecke operatorsMathematical Proceedings of the Cambridge Philosophical Society, 88
- Publisher
- Springer Journals
- Copyright
- Copyright © 2007 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Mathematics, general; Mathematical and Computational Physics
- ISSN
- 1678-7544
- eISSN
- 1678-7714
- DOI
- 10.1007/s00574-007-0042-5
- Publisher site
- See Article on Publisher Site
Abstract
For integers a, b and n > 0, define $$ A_{\Gamma } {\left( {a,b,n} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{r = 1}} \\ {{n\nmid b}} \\ \end{array} }^n {'\;\;{\left( {{\left( {\frac{{ar}} {n}} \right)}} \right)}} }\ln \Gamma {\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)} $$ and $$ B_{\Gamma } {\left( {a,b,n} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{r = 1}} \\ {{n\nmid b}} \\ \end{array} }^n {'\;\;{\left( {{\left( {\frac{{ar}} {n}} \right)}} \right)}} }\frac{{{\Gamma }\ifmmode{'}\else$'$\fi{\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)}}} {{\Gamma {\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)}}}, $$ where $$ {\sum\limits{_r} ' } $$ denotes the summation over all r such that (r, n) = 1, and $$ \overline{r} $$ is defined by the equation $$ r\overline{r} \equiv 1\;\bmod n $$ . The two sums are analogous to the homogeneous Dedekind sum S(a,b, n). The functional equations for A Γ and B Γ are established. Furthermore, Knopp's identity on Dedekind sum is extended.
Journal
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Jan 1, 2007