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 } − 1 2 if x �∈ Z (( x )) := if x ∈ Z . 0 ( { x } = x − [ x ] = fractional part of x .) For a, b ∈ N := { n ∈ Z : n > 0 } , we define the Dedekind sum as �� �� �� �� � ka k s ( a, b ) := b b k mod b � 1 cot πka cot πk = b . 4 b b k mod 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 B k ( x ) are de- fined through ze xz � B k ( x ) z k . e z − 1 = k ! k ≥ 0 2 , B 2 ( x ) = x 2 − x + 1 ( B 1 ( x ) = x − 1 6 , etc.) The Bernoulli numbers are B k := B k (0) . The Bernoulli functions B k ( x ) are the peri- odized Bernoulli polynomials: B k ( x ) := B k ( { x } ) . Apostol (1950’s) introduced � � � k ka b B n , b k mod b generalized by Carlitz and Mikol´ as to � � � � � kb kc s m,n ( a ; b, c ) := B m B n . a a k mod b 3

For a, b ∈ N , x, y ∈ R , the Dedekind- Rademacher sum (1960’s) is defined by s ( a, b ; x, y ) := �� �� �� k + y �� � a k + y − x . b b k mod b (A special version of this sum had been de- fined earlier by Meyer and Dieter.) Tak´ acs (1970’s) introduced � k + y � � � � a k + y B 1 B n − x , b b k mod b further generalized by Halbritter (1980’s) and Hall, Wilson, and Zagier (1990’s) to � a b c � := s m,n x y z � � � � � b k + x c k + x B m − y B n − z . a a k mod a 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 a k + z b k + z cot π − x cot π − y . c c c k mod c These include as special cases various mod- ified Dedekind sums introduced by Berndt, such as b ( − 1) k +[ ak/b ] �� �� � k . b k =1 Finally, Zagier (1970’s) introduced s ( a 0 ; a 1 , . . . , a d ) := a 0 − 1 � ( − 1) d/ 2 a 0 · · · cot πka d cot πka 1 . a 0 a 0 k =1 5

Definition. For a 0 , . . . , a d ∈ N , m 0 , . . . , m d ∈ N 0 := N ∪ { 0 } , z 0 , . . . , z d ∈ C , we define the Dedekind cotangent sum as   a 0 a 1 ··· a d  :=  m 0 m 1 ··· m d c z 0 z 1 ··· z d d � � � � cot ( m j ) π k + z 0 1 a j a 0 − z j . a m 0+1 0 j =1 k mod a 0 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 , � � B m n B m = ( − p ) m + p � m p − 1 � cot ( m − 1) � � � e 2 πkn/p . i πk m 2 p p k =1 6

Corollary. For a, b, c ∈ N pairwise relatively prime, � � � � � s m,n ( a ; b, c ) def kb kc = B m B n a a k mod a a m + n − 1 + mn ( − 1) ( m − n ) / 2 = B m B n 2 m + n a m + n − 1 · a − 1 cot ( m − 1) � � cot ( n − 1) � � � πkc πkb · a a k =1   a b c = mn ( − 1) ( m − n ) / 2 def   m + n − 2 n − 1 m − 1 c 2 m + n 0 0 0 + B m B n a m + n − 1 . Another note: a − 1 � � �� �� � k kb B m a a k =1 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 cot πka cot πk 4 b b b k mod b Theorem (Dedekind). If a, b ∈ N are rela- tively prime then � � s ( a, b ) + s ( b, a ) = − 1 4 + 1 a b + 1 ab + b 12 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 . 10

c ( a, b, c ; x, y, z ) := � � � � � 1 a k + z b k + z cot π − x cot π − y . c c c k mod c 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 . 11

s ( a 0 ; a 1 , . . . , a d ) := a 0 − 1 � ( − 1) d/ 2 a 0 · · · cot πka d cot πka 1 . a 0 a 0 k =1 Theorem (Zagier). If a 0 , . . . , a d ∈ N are pairwise relatively prime then d � s ( a n ; a 0 , . . . , � a n , . . . , a d ) n =0 = something simple . ”Proof”: Integrate f ( z ) = cot πa 0 z · · · cot πa d z . along γ . 12

  a 0 a 1 ··· a d   := m 0 m 1 ··· m d c z 0 z 1 ··· z d d � � � � cot ( m j ) π k + z 0 1 a j a 0 − z j . a m 0+1 0 k mod a 0 j =1 Theorem. Let a 0 , . . . , a d ∈ N , m 0 , . . . m d ∈ N 0 , z 0 , . . . , z d ∈ C . Then d 0 ··· � n ··· a ld � � a l 0 a ln ( − 1) m n m n ! d l n ! ··· l d ! · l 0 ! ··· � n =0 l 0 ,..., � ln,...,ld ≥ 0 l 0+ ··· + � ln + ··· + ld = mn   · · · · · · a n a 0 a n a d �   � · c m n m 0 + l 0 · · · m n + l n · · · m d + l d · · · · · · z n z 0 z n z d � � ( − 1) d/ 2 if all m k = 0 and d is even = 0 otherwise, 13

if for all distinct i, j ∈ { 0 , . . . , d } and all m, n ∈ Z , − n + z j m + z i �∈ Z . a i a j ”Proof”: Integrate d � � � cot ( m j ) π f ( z ) = a j z − z j j =0 along γ . 14

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 � � � n � d b + ka, ad = σ ( n ) s ( b, a ) . s k mod d d | n 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 ( m 1 , m 2 ) . 16

Definition. Let a, a 1 , . . . , a d ∈ N . � ka 1 � � ka d � � S ( a ; a 1 , . . . , a d ) := f 1 · · · f d a a k mod a is called of Dedekind type with weight ( m 1 , . . . , m d ) if for all j = 1 , . . . , d , f j ( x + 1) = f j ( x ) and for all a ∈ N , � � � x + k = a m j f j ( ax ) . f j a k mod a Note that the Bernoulli functions B m ( 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 17

Theorem. Let n, a, a 1 , . . . , a d ∈ N . If � ka 1 � � ka d � � S ( a ; a 1 , . . . , a d ) := f 1 · · · f d a a k mod a is of Dedekind type with weight ( m 1 , . . . , m d ) then � � b − m 1 −···− m d r 1 ,...,r d mod b b | n � � ab ; n b a 1 + r 1 a , . . . , n S b a d + r d a = n σ d − 1 − m 1 −···− m d ( n ) S ( a ; a 1 , . . . , a d ) . Here σ m ( n ) := � d | n d m . Our proof is along the exact same lines as Zheng’s proof for d = 2 , a slick application of the M¨ obius µ -function. 18

Corollary. For n, a 0 , . . . , a d ∈ N , m 0 , . . . , m d ∈ N 0 , � � b m 0 +1 − m 1 −···− m d − d r 1 ,...,r d mod b b | n   a 0 b n ba 1 + r 1 a 0 · · · n ba d + r d a 0   · · · m 0 m 1 m d c · · · 0 0 0   a 1 · · · a 0 a d  .  = n σ − m 1 −···− m d − 1 ( n ) c m 0 m 1 · · · m d 0 · · · 0 0 Corollary. For n, a 0 , . . . , a d ∈ N , � � b 1 − d r 1 ,...,r d mod b b | n s ( a 0 b ; n b a 1 + r 1 a 0 , . . . , n b a d + r d a 0 ) = σ ( n ) s ( a 0 ; a 1 , . . . , a d ) . 19