Dedekind cotangent sums Matthias Beck State University of New York - - PDF document

Dedekind cotangent sums Matthias Beck State University of New York Binghamton www.math.binghamton.edu/matthias

Define the sawtooth function ((x)) by ((x)) :=

• {x} − 1

2 if x ∈ Z

if x ∈ Z . ({x} = x − [x] = fractional part of x .) For a, b ∈ N := {n ∈ Z : n > 0} , we define the Dedekind sum as s(a, b) :=

• k mod b
• ka

b k b

• =

1 4b

• k mod b

cot πka

b

cot πk

b .

Since their introduction by Dedekind in the 1880’s, these sums and their generalizations have appeared in various areas such as ana- lytic and algebraic number theory, topology, algebraic and combinatorial geometry, and algorithmic complexity.

2

The Bernoulli polynomials Bk(x) are de- fined through zexz ez − 1 =

• k≥0

Bk(x) k! zk . (B1(x) = x − 1

2, B2(x) = x2 − x + 1 6, etc.)

The Bernoulli numbers are Bk := Bk(0) . The Bernoulli functions Bk(x) are the peri-

• dized Bernoulli polynomials:

Bk(x) := Bk({x}) . Apostol (1950’s) introduced

• k mod b

k b Bn

• ka

b

• ,

generalized by Carlitz and Mikol´ as to sm,n(a; b, c) :=

• k mod b

Bm

• kb

a

• Bn
• kc

a

• .

3

For a, b ∈ N , x, y ∈ R , the Dedekind- Rademacher sum (1960’s) is defined by s(a, b; x, y) :=

• k mod b
• a k+y

b

− x k+y

b

• .

(A special version of this sum had been de- fined earlier by Meyer and Dieter.) Tak´ acs (1970’s) introduced

• k mod b

B1 k+y

b

• Bn
• a k+y

b

− x

• ,

further generalized by Halbritter (1980’s) and Hall, Wilson, and Zagier (1990’s) to sm,n a b c x y z

• :=
• k mod a

Bm

• b k+x

a

− y

• Bn
• c k+x

a

− z

• .

4

Around 1980, Meyer and Sczech, and Di- eter independently introduced the cotangent sum, defined for a, b ∈ N , x, y ∈ R by c(a, b, c; x, y, z) :=

1 c

• k mod c

cot π

• a k+z

c

− x

• cot π
• b k+z

c

− y

• .

These include as special cases various mod- ified Dedekind sums introduced by Berndt, such as

b

• k=1

(−1)k+[ak/b]

k b

• .

Finally, Zagier (1970’s) introduced s(a0; a1, . . . , ad) :=

(−1)d/2 a0 a0−1

• k=1

cot πka1

a0 · · · cot πkad a0

.

5

• Definition. For a0, . . . , ad ∈ N , m0, . . . , md ∈

N0 := N ∪ {0} , z0, . . . , zd ∈ C , we define the Dedekind cotangent sum as c  

a0 a1 ··· ad m0 m1 ··· md z0 z1 ··· zd

  :=

1 am0+1

• k mod a0

d

• j=1

cot(mj) π

• aj

k+z0 a0 − zj

• .

Reason for introducing cotangent derivatives:

• they appear in lattice point enumeration

formulas for polyhedra (Diaz-Robins)

• they are essentially the discrete Fourier

transforms of the Bernoulli functions: for m ≥ 2 , Bm

• n

p

• =

Bm (−p)m +

m

• i

2p

m p−1

• k=1

cot(m−1)

πk p

• e2πkn/p .

6

• Corollary. For a, b, c ∈ N pairwise relatively

prime, sm,n(a; b, c) def =

• k mod a

Bm

• kb

a

• Bn
• kc

a

• = BmBn

am+n−1 + mn (−1)(m−n)/2 2m+nam+n−1 ·

·

a−1

• k=1

cot(m−1)

πkc a

• cot(n−1)

πkb a

• def

= mn (−1)(m−n)/2

2m+n

c  

a b c m+n−2 n−1 m−1

  + BmBn

am+n−1 .

Another note:

a−1

• k=1

Bm

• k

a kb a

• appears naturally in the study of plane par-

tition enumeration (Almkvist, 1990’s).

7

Three themes for Dedekind sums:

• 1. Reciprocity laws
• 2. Petersson-Knopp identities
• 3. Computability properties

8

1. “If you had done something twice, you are likely to do it again.” Brian Kernighan and Bob Pike (The Unix Programming Environment, Pren- tice Hall, p. 97)

9

s(a, b) := 1

4b

• k mod b

cot πka

b

cot πk

b

Theorem (Dedekind). If a, b ∈ N are rela- tively prime then s(a, b) + s(b, a) = −1

4 + 1 12

• a

b + 1 ab + b a

• = something simple .

”Proof” (Rademacher? Carlitz?): Integrate the function f(z) = cot(πaz) cot(πbz) cot(πz) along γ = [x+iy, x−iy, x+1−iy, x+1+iy, x+iy] , for suitably chosen x and y .

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c(a, b, c; x, y, z) :=

1 c

• k mod c

cot π

• a k+z

c

− x

• cot π
• b k+z

c

− y

• .

Theorem (Dieter). Let a, b, c ∈ N be pair- wise relatively prime. Then c (a, b, c; x, y, z) + c (b, c, a; y, z, x) +c (c, a, b; z, x, y) = something simple . ”Proof”: Integrate f(w) = cot π(aw−x) cot π(bw−y) cot π(cw−z) along γ .

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s(a0; a1, . . . , ad) :=

(−1)d/2 a0 a0−1

• k=1

cot πka1

a0 · · · cot πkad a0

. Theorem (Zagier). If a0, . . . , ad ∈ N are pairwise relatively prime then

d

• n=0

s (an; a0, . . . , an, . . . , ad) = something simple . ”Proof”: Integrate f(z) = cot πa0z · · · cot πadz . along γ .

12

c  

a0 a1 ··· ad m0 m1 ··· md z0 z1 ··· zd

  :=

1 am0+1

• k mod a0

d

• j=1

cot(mj) π

• aj

k+z0 a0 − zj

• .
• Theorem. Let a0, . . . , ad ∈ N , m0, . . . md ∈

N0 , z0, . . . , zd ∈ C . Then

d

• n=0

(−1)mnmn!

• l0,...,

ln,...,ld≥0 l0+···+ ln+···+ld=mn

al0

0 ···

aln

n ···ald d

l0!··· ln!···ld! ·

· c  

an a0

· · ·

• an

· · ·

ad mn m0+l0 · · ·

• mn+ln · · · md+ld

zn z0

· · ·

• zn

· · ·

zd

  =

• (−1)d/2 if all mk = 0 and d is even
• therwise,

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if for all distinct i, j ∈ {0, . . . , d} and all m, n ∈ Z ,

m+zi ai

− n+zj

aj

∈ Z . ”Proof”: Integrate f(z) =

d

• j=0

cot(mj) π

• ajz − zj
• along γ .

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2. ”Mathematics is a collection of cheap tricks and dirty jokes.” Lipman Bers

15

Theorem (Petersson-Knopp). If a, b ∈ N are relatively prime then

• d|n
• k mod d

s n

db + ka, ad

• = σ(n) s(b, a) .

This identity was stated by Petersson in the 1970’s (with additional congruence restric- tions on a and b ) and proved by Knopp in

• 1980. For n prime, the identity was already

known to Dedekind. It was generalized by Parson and Rosen to Apostol’s generalized Dedekind sums, by Apos- tel and Vu to their ’sums of the Dedekind type’ (both 1980’s), and, most broadly, by Zheng (1990’s) to what we will call sums of Dedekind type with weight (m1, m2) .

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• Definition. Let a, a1, . . . , ad ∈ N .

S (a; a1, . . . , ad) :=

• k mod a

f1 ka1

a

• · · · fd

kad

a

• is called of Dedekind type with weight

(m1, . . . , md) if for all j = 1, . . . , d , fj(x + 1) = fj(x) and for all a ∈ N ,

• k mod a

fj

• x + k

a

• = amj fj(ax) .

Note that the Bernoulli functions Bm(x) satisfy this identity (with ’weight’ −m + 1 ), as do the functions cot(m)(πx) (with ’weight’ m + 1 ). Zheng’s theorem is the ’two-dimensional’ case of the following

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• Theorem. Let n, a, a1, . . . , ad ∈ N . If

S (a; a1, . . . , ad) :=

• k mod a

f1 ka1

a

• · · · fd

kad

a

• is of Dedekind type with weight (m1, . . . , md)

then

• b|n

b−m1−···−md

• r1,...,rd mod b

S

• ab; n

ba1 + r1a , . . . , n bad + rda

• = n σd−1−m1−···−md(n) S (a; a1, . . . , ad) .

Here σm(n) :=

d|n dm .

Our proof is along the exact same lines as Zheng’s proof for d = 2 , a slick application

• f the M¨
• bius µ-function.

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• Corollary. For n, a0, . . . , ad ∈ N, m0, . . . , md ∈

N0 ,

• b|n

bm0+1−m1−···−md−d

• r1,...,rd mod b

c  

a0b n ba1+r1a0 · · · n bad+rda0 m0 m1

· · ·

md

· · ·   = n σ−m1−···−md−1(n) c  

a0 a1 · · · ad m0 m1 · · · md 0 · · ·

  .

• Corollary. For n, a0, . . . , ad ∈ N ,
• b|n

b1−d

• r1,...,rd mod b

s(a0b; n

ba1 + r1a0, . . . , n bad + rda0)

= σ(n) s(a0; a1, . . . , ad) .

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