(pp. 26-44)

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 of$s(h,k)$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,...