Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" TY - CHAP TI - ARITHMETIC PROPERTIES OF THE DEDEKIND SUMS A2 - RADEMACHER, HANS A2 - GROSSWALD, EMIL AB -

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,

EP - 44 ET - 1 PB - Mathematical Association of America PY - 1972 SN - 9780883850480 SP - 26 T2 - Dedekind Sums UR - http://www.jstor.org/stable/10.4169/j.ctt5hh823.7 VL - 16 Y2 - 2021/08/05/ ER -