@inbook{10.4169/j.ctt5hh823.7, ISBN = {9780883850480}, URL = {http://www.jstor.org/stable/10.4169/j.ctt5hh823.7}, abstract = {The reciprocity law of the Dedekind sums always contains two (and in some generalizations three and even more) Dedekind sums. We focus our attention now on a single Dedekind sum, its properties and its connections with other mathematical topics.Since$(( - x)) = - ((x))$it is clear that(33a)$s( - h,k) = - s(h,k)$and(33b)$s(h, - k) = s(h,k)$.If we define${h'}$by$hh' \equiv 1(\bmod k)$, then(33c)$s(h',k) = s(h,k)$Indeed,$\sum\limits_{\mu \bmod k} {\left( {\left( {\frac{\mu }{k}} \right)} \right)} \left( {\left( {\frac{{h\mu }}{k}} \right)} \right) = \sum\limits_{\mu \bmod k} {\left( {\left( {\frac{{h'\mu }}{k}} \right)} \right)} \left( {\left( {\frac{{hh'\mu }}{k}} \right)} \right)$$= \sum\limits_{\mu \bmod k} {\left( {\left( {\frac{{h'\mu }}{k}} \right)} \right)} \left( {\left( {\frac{\mu }{k}} \right)} \right) = s(h',k).Next, we may state the following:Theorem 2. The denominator ofs(h,k)is a divisor of 2k(3,k).Proof of Theorem 2.s(h,k) = \sum\limits_{\mu = 1}^{k - 1} {\left( {\frac{\mu }{k} - \left[ {\frac{\mu }{k}} \right] - \frac{1}{2}} \right)} \left( {\frac{{h\mu }}{k} - \left[ {\frac{{h\mu }}{k}} \right] - \frac{1}{2}} \right)$$ = \frac{h}{{{k^2}}}\sum\limits_{\mu = 1}^{k - 1} {{\mu ^2}} + \frac{A}{{2k}} + \frac{1}{4}\sum\limits_{\mu = 1}^{k - 1} 1$If k is even, 4 | 2k; if is odd, fc—1 is even, and the last fraction has, after reduction,}, bookauthor = {HANS RADEMACHER and EMIL GROSSWALD}, booktitle = {Dedekind Sums}, edition = {1}, pages = {26--44}, publisher = {Mathematical Association of America}, title = {ARITHMETIC PROPERTIES OF THE DEDEKIND SUMS}, volume = {16}, year = {1972} }