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Dedekind sums Mirco Kraenz The Dedekind sums ingredients Dedekind sums Fourier- Dedekind sums Restricted partition Mirco Kraenz function Reciprocity laws Seminar Discrete Convex Geometry How to Seminar Chair: Prof. Dr. Martin Henk compute Technical University Berlin Dedekind sums Mordell- February 7, 2017 Pommersheim tetrahedron

Outline Dedekind sums Mirco Kraenz 1 The ingredients The Dedekind sums ingredients Dedekind sums Fourier-Dedekind sums Fourier- Dedekind Restricted partition function sums Restricted partition function Reciprocity 2 Reciprocity laws laws How to compute Dedekind 3 How to compute Dedekind sums sums Mordell- Pommersheim 4 Mordell-Pommersheim tetrahedron tetrahedron

Dedekind sums - Definition Dedekind sums Mirco Kraenz The ingredients Dedekind sums Let a , b relatively prime integers, b > 0. Define the Dedekind Fourier- Dedekind sums sums Restricted b − 1 partition �� ka ���� k function �� � s ( a , b ) = , Reciprocity b b laws k =1 How to where (( x )) denotes the saw-tooth function. compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties Dedekind sums Mirco Kraenz The ingredients Dedekind sums b − 1 �� ka ���� k Fourier- �� � Dedekind s ( a , b ) = sums b b Restricted partition k =1 function Reciprocity laws periodic in a with period b How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties Dedekind sums Mirco Kraenz The ingredients Dedekind sums b − 1 �� ka ���� k Fourier- �� � Dedekind s ( a , b ) = sums b b Restricted partition k =1 function Reciprocity laws periodic in a with period b How to special values s (1 , k ) = − 1 4 + 1 6 k + k compute 12 Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties Dedekind sums Mirco Kraenz The ingredients Dedekind sums b − 1 �� ka ���� k Fourier- �� � Dedekind s ( a , b ) = sums b b Restricted partition k =1 function Reciprocity laws periodic in a with period b How to special values s (1 , k ) = − 1 4 + 1 6 k + k compute 12 Dedekind sums our first goal: s ( a , b ) + s ( b , a ) = some simple function Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Definition Dedekind sums Mirco Kraenz The ingredients Let a 1 , a 2 , . . . , a d pairwise relatively prime, positive integers. Dedekind sums Fourier- Let n some integer. Define the Fourier-Dedekind sums Dedekind sums Restricted partition b − 1 function ξ nk s n ( a 1 , . . . , a d ; b ) = 1 � b , Reciprocity i =1 (1 − ξ ka i laws b Π d b ) k =1 How to compute where ξ b = e 2 π i / b denotes the b -th root of unity. Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties Dedekind sums Mirco Kraenz The ingredients b − 1 ξ nk s n ( a 1 , . . . , a d ; b ) = 1 Dedekind sums � b Fourier- Dedekind i =1 (1 − ξ ka i b Π d b ) sums k =1 Restricted partition function Reciprocity commutative in a i arguments laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties Dedekind sums Mirco Kraenz The ingredients b − 1 ξ nk s n ( a 1 , . . . , a d ; b ) = 1 Dedekind sums � b Fourier- Dedekind i =1 (1 − ξ ka i b Π d b ) sums k =1 Restricted partition function Reciprocity commutative in a i arguments laws → write s n ( A ; b ) with A = { a 1 , a 2 . . . a d } How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties Dedekind sums Mirco Kraenz The ingredients b − 1 ξ nk s n ( a 1 , . . . , a d ; b ) = 1 Dedekind sums � b Fourier- Dedekind i =1 (1 − ξ ka i b Π d b ) sums k =1 Restricted partition function Reciprocity commutative in a i arguments laws → write s n ( A ; b ) with A = { a 1 , a 2 . . . a d } How to compute periodic in each a i with period b Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties Dedekind sums Mirco Kraenz The ingredients b − 1 ξ nk s n ( a 1 , . . . , a d ; b ) = 1 Dedekind sums � b Fourier- Dedekind i =1 (1 − ξ ka i b Π d b ) sums k =1 Restricted partition function Reciprocity commutative in a i arguments laws → write s n ( A ; b ) with A = { a 1 , a 2 . . . a d } How to compute periodic in each a i with period b Dedekind sums s 0 ( a , 1; b ) = − s ( a , b ) + b − 1 Mordell- 4 b Pommersheim tetrahedron

Restricted partition function Dedekind sums Mirco Kraenz The ingredients Let A = { a 1 , a 2 . . . a d } ⊂ N d , where the a i are pairwise Dedekind sums Fourier- relatively prime. For a positive integer n , the restricted Dedekind sums partition function is defined to be Restricted partition function Reciprocity d laws � p A ( n ) = # { ( k 1 . . . k d ) ∈ Z d ≥ 0 | k i a i = n } . How to compute i =1 Dedekind sums Mordell- Pommersheim tetrahedron

Polynomial part of p A ( n ) Dedekind sums Mirco Kraenz d d The � � s − n ( ˆ ( − 1) i B i + ingredients p A ( n ) = A i ; a i ) Dedekind sums Fourier- i =1 i =1 Dedekind sums with B i coefficients at z = 1 of the partial fraction expansion of Restricted partition function the shifted generating function of p A ( n ) . B i are polynomials in Reciprocity n . Here ˆ A i = A \ { a i } . laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Polynomial part of p A ( n ) Dedekind sums Mirco Kraenz d d The � � s − n ( ˆ ( − 1) i B i + ingredients p A ( n ) = A i ; a i ) Dedekind sums Fourier- i =1 i =1 Dedekind sums with B i coefficients at z = 1 of the partial fraction expansion of Restricted partition function the shifted generating function of p A ( n ) . B i are polynomials in Reciprocity n . Here ˆ A i = A \ { a i } . Therefore, define laws How to compute d Dedekind � ( − 1) i B i poly A ( n ) = sums Mordell- i =1 Pommersheim tetrahedron the polynomial part of the restricted partition function.

Zagier reciprocity Dedekind sums Mirco Kraenz The ingredients Theorem 8.4 (Zagier reciprocity) Dedekind sums Fourier- For A = { a 1 , a 2 . . . a d } ⊂ N d , with the a i are pairwise relatively Dedekind sums prime, Restricted partition function d � s 0 ( ˆ Reciprocity A i ; a i ) = 1 − poly A (0) , laws i =1 How to compute where ˆ Dedekind A i = A \ { a i } . sums Mordell- Pommersheim tetrahedron

Dedekind’s reciprocity law Dedekind sums Mirco Kraenz The ingredients Dedekind sums Corollary 8.5 (Dedekind’s reciprocity law) Fourier- Dedekind sums For all a , b relatively prime, positive integers we have Restricted partition function s ( a , b ) + s ( b , a ) = 1 a + 1 ab ) − 1 12( a b + b Reciprocity 4 . laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Equalities for Dedekind sums Dedekind sums Mirco Kraenz periodicity The ingredients s ( a , b ) = s ( a mod b , b ) Dedekind sums Fourier- Dedekind Dedekind’s reciprocity law sums Restricted partition function s ( a , b ) + s ( b , a ) = 1 a + 1 ab ) − 1 12( a b + b 4 . Reciprocity laws How to special values compute Dedekind sums s (1 , k ) = − 1 4 + 1 6 k + k Mordell- Pommersheim 12 tetrahedron

Mordell-Pommersheim tetrahedron Dedekind sums Mirco Kraenz The ingredients Dedekind sums Fourier- For positive integers a , b , c call Dedekind sums Restricted partition ≥ 0 | x a + y b + z � � ( x , y , z ) ∈ R 3 function P = c ≤ 1 Reciprocity laws the Mordell-Pommersheim tetrahedron . How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron Dedekind sums For positive integers a , b , c call Mirco Kraenz ≥ 0 | x a + y b + z � � ( x , y , z ) ∈ R 3 The tP = c ≤ t ingredients Dedekind sums Fourier- the t-th dilate of the Mordell-Pommersheim tetrahedron . Dedekind sums Restricted partition function Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron Dedekind sums For positive integers a , b , c call Mirco Kraenz ≥ 0 | x a + y b + z � � ( x , y , z ) ∈ R 3 The tP = c ≤ t ingredients Dedekind sums Fourier- the t-th dilate of the Mordell-Pommersheim tetrahedron . Dedekind sums Restricted Its Lattice-point enumerator is partition function Reciprocity ≥ 0 | k a + l b + m � � laws ( k , l , m ) ∈ Z 3 L P ( t ) = # c ≤ t How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron Dedekind sums For positive integers a , b , c call Mirco Kraenz ≥ 0 | x a + y b + z � � ( x , y , z ) ∈ R 3 The tP = c ≤ t ingredients Dedekind sums Fourier- the t-th dilate of the Mordell-Pommersheim tetrahedron . Dedekind sums Restricted Its Lattice-point enumerator is partition function Reciprocity ≥ 0 | k a + l b + m � � laws ( k , l , m ) ∈ Z 3 L P ( t ) = # c ≤ t How to compute � � ( k , l , m ) ∈ Z 3 Dedekind = # ≥ 0 | kbc + lac + mab ≤ abct sums Mordell- Pommersheim tetrahedron