# Dedekind Sums

EMIL GROSSWALD
Volume: 16
Edition: 1
Pages: 119
https://www.jstor.org/stable/10.4169/j.ctt5hh823

1. Front Matter
(pp. i-vi)
2. INTRODUCTION
(pp. vii-viii)
Ivan Niven

Professor Hans Rademacher was invited by the Mathematical Association of America to deliver the Earle Raymond Hedrick lectures at the 1963 summer meeting in Boulder, Colorado. Professor Rademacher chose the topic Dedekind Sums, and prepared a set of notes from which to deliver the lectures. However, a temporary illness prevented him from giving the lectures, and he prevailed upon his colleague and former student, Emil Grosswald, to make the presentation from the notes.

Professor Rademacher never edited for publication the notes he had prepared for the Hedrick lectures. However, after his death in 1969, the manuscript was found among his...

3. PREFACE
(pp. ix-xiv)
Emil Grosswald
(pp. xv-xvi)
5. CHAPTER 1 INTRODUCTION
(pp. 1-3)

At first glance, the Dedekind sums seem to be a highly specialized subject. These sums denoted by$s(h,k)$are defined as follows: Leth, kbe integers,$(h,k) = 1$,$k \ge 1$then we set

(1)$s(h,k) = \sum\limits_{\mu = 1}^k {\left( {\frac{{h\mu }}{k}} \right)} \left( {\frac{\mu }{k}} \right)$

Here and in the following the symbol ((x)) is defined by

(2)$((x)) = \left\{ \begin{array}{l} x - [x] - 1/2 \\ 0 \\ \end{array} \right.$ifxis not an integer, ifxis an integer,

with [x] the greatest integer not exceedingx. This is the well-known sawtooth function of period 1 (see Figure 1),

which at the points of discontinuity takes the mean value between the...

6. CHAPTER 2 SOME PROOFS OF THE RECIPROCITY FORMULA
(pp. 4-25)

Theorem 1. (Reciprocity Theorem).Let h and k be two coprime integers. Then

(4)$s(h,k) + s(k,h) = - \frac{1}{4} + \frac{1}{{12}}\left( {\frac{h}{k} + \frac{1}{{hk}} + \frac{k}{h}} \right)$.

A. First Proof. Let us begin with a proof which is a variant of one given by Ulrich Dieter [16]. We need first the

Lemma 1.$s(h,k) + s(k,h) = - \frac{1}{4} + \frac{1}{{12}}\left( {\frac{h}{k} + \frac{1}{{hk}} + \frac{k}{h}} \right)$

Proof: We consider the difference

$D(x) = \sum\limits_{\lambda \bmod k} {\left( {\left( {\frac{{\lambda + x}}{k}} \right)} \right) - ((x))}$

This function is obviously periodic inxwith period 1. We may thus restrict our discussion to the range$0 \le x < 1$. If we choose, in particular, the residue system$\lambda = 0,1, \cdots ,k - 1$, we have

$D\left( 0 \right) = \sum\limits_{\lambda = 1}^{k - 1} {\left( {\frac{\lambda }{k} - \frac{1}{2}} \right)} = \frac{{k - 1}}{2} - \frac{{k - 1}}{2} = 0$

Similarly, for$0 < x < 1$,

$D(x) = \sum\limits_{\lambda = 0}^{k - 1} {\left( {\frac{\lambda }{k} + \frac{x}{k} - \frac{1}{2}} \right)} - \left( {x - \frac{1}{2}} \right)$

$= \frac{{k - 1}}{2} + x - \frac{k}{2} - x + \frac{1}{2} = 0$.

This shows that$D(x) = 0$for all values ofx, and completes the proof of the Lemma.

Lemma 2.

(5)...

7. CHAPTER 3 ARITHMETIC PROPERTIES OF THE DEDEKIND SUMS
(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,...

8. CHAPTER 4 DEDEKIND SUMS AND MODULAR TRANSFORMATIONS
(pp. 45-63)

A. Then-Function and the Function Φ(M) . The functionn(τ) defined in Chapter 1 (see (3)) and now generally known as the Dedekindn-function, appears already in the work of Jacobi and Weierstrass on elliptic functions in the form

(54)$\Delta (\tau ) = Cn{(\tau )^{24}}$.

HereCis an unimportant numerical constant. Actually, if one uses the proper normalization (see, e.g. [31], p. 108), thenC= 1. This function has to do with the pattern of periods of the elliptic functions represented by the point lattice

$\Omega = \left\{ {{m_1}{\omega _1} + {m_2}{\omega _2}} \right\}$.

Here${\omega _1}$and${\omega _2}$are two generators of the point lattice and${m_1},{m_2}$run independently through...

9. CHAPTER 5 GENERALIZATIONS
(pp. 64-65)

We shall close this presentation of Dedekind sums with the mention of some of their generalizations. There are several in which the function ((x)), which is essentially the first Bernoulli polynomial${B_1}(y) = y - 1/2$of$y = x - [x]$, is replaced by higher Bernoulli polynomials. They play roles in special problems of partitions.

In the work of C. Meyer [32], [33], but also already in some investigations by J. Lehner [29] and J. Livingood [30], certain Dedekind sums appear in which the summandμis restricted by congruence conditions. These types of Dedekind sums are in full generality contained in the definition

$s(h,k;x,y) = \sum\limits_{\mu \bmod k} {\left( {\left( {\frac{{\mu + y}}{k}} \right)} \right)} \left( {\left( {h\frac{{\mu + y}}{k} + x} \right)} \right)$

wherexandy...

10. CHAPTER 6 SOME REMARKS ON THE HISTORY OF THE DEDEKIND SUMS
(pp. 66-80)

Bernhard Riemann died on July 20, 1866 at the age of forty. According to his wish, his manuscripts, notes, etc., were entrusted to R. Dedekind. It turned out that this was a rather mixed lot; there were some practically finished papers, then some drafts of varying degrees of completeness, and some that were just fragmentary sketches.

Among the latter were two notes related to the theory of elliptic modular function as presented in Jacobi’sFundamenta Nova.In Jacobi’s work, the parameterqsatisfies|q|< 1, while in these notes, Riemann considers the limiting case|q|= 1. The first...

11. APPENDIX
(pp. 81-93)
12. REFERENCES
(pp. 94-97)
13. LIST OF THEOREMS AND LEMMAS
(pp. 98-98)
14. NAME INDEX
(pp. 99-99)
15. SUBJECT INDEX
(pp. 100-102)