Relations for Barnes Zeta Functions Abdelmejid Bayad Universit e - PowerPoint PPT Presentation

Relations for Barnes Zeta Functions Abdelmejid Bayad Universit´ e d’Evry Val d’Essonne Matthias Beck San Francisco State University math.sfsu.edu/beck In fond memory of my teacher, mentor, and friend Marvin Knopp

Bernoulli Relations � n � Euler n � � z k z e z − 1 = B k − → B j B n − j = − n B n − 1 − ( n − 1) B n k ! j et al k ≥ 0 j =0 Relations for Barnes Zeta Functions Matthias Beck 3

Bernoulli Relations � n � Euler n � � z k z e z − 1 = B k − → B j B n − j = − n B n − 1 − ( n − 1) B n k ! j et al k ≥ 0 j =0 N¨ orlund (1922): Relations for Bernoulli polynomials B k ( x ) defined through � z e xz B k ( x ) z k e z − 1 = k ! k ≥ 0 Relations for Barnes Zeta Functions Matthias Beck 3

Bernoulli Relations � n � Euler n � � z k z e z − 1 = B k − → B j B n − j = − n B n − 1 − ( n − 1) B n k ! j et al k ≥ 0 j =0 N¨ orlund (1922): Relations for Bernoulli polynomials B k ( x ) defined through � z e xz B k ( x ) z k e z − 1 = k ! k ≥ 0 Dilcher (1996): Relations for Bernoulli numbers of order n defined through � � n � z k z B ( n ) = e z − 1 k k ! k ≥ 0 and their polynomial generalization. Relations for Barnes Zeta Functions Matthias Beck 3

Bernoulli Relations � n � n Euler � � z k z e z − 1 = − → B j B n − j = − n B n − 1 − ( n − 1) B n B k k ! j et al j =0 k ≥ 0 N¨ orlund (1922): Relations for Bernoulli polynomials B k ( x ) defined through � z e xz B k ( x ) z k e z − 1 = k ! k ≥ 0 Dilcher (1996): Relations for Bernoulli numbers of order n defined through � � n � z k z B ( n ) = e z − 1 k k ! k ≥ 0 and their polynomial generalization. Goal: Relations for Bernoulli–Barnes numbers B k ( a ) defined through � z n B k ( a ) z k a = ( a 1 , a 2 , . . . , a n ) ∈ R n ( e a 1 z − 1) · · · ( e a n z − 1) = k ! , > 0 k ≥ 0 Relations for Barnes Zeta Functions Matthias Beck 3

Bernoulli–Barnes Relations � z n B k ( a ) z k a = ( a 1 , a 2 , . . . , a n ) ∈ R n ( e a 1 z − 1) · · · ( e a n z − 1) = k ! , > 0 k ≥ 0 Theorem 1 For n ≥ 3 and odd m ≥ 1 � � n + j − 4 � n � � 1 1 if n = m = 3 2 B m − n + j ( a I ) = j − 2 ( m − n + j )! 0 otherwise j = n − m | I | = j where the inner sum is over all subsets I ⊆ { 1 , 2 , . . . , n } of cardinality j and a I := ( a i : i ∈ I ) . Relations for Barnes Zeta Functions Matthias Beck 4

Bernoulli–Barnes Relations � z n B k ( a ) z k a = ( a 1 , a 2 , . . . , a n ) ∈ R n ( e a 1 z − 1) · · · ( e a n z − 1) = k ! , > 0 k ≥ 0 Theorem 1 For n ≥ 3 and odd m ≥ 1 � � n + j − 4 � n � � 1 1 if n = m = 3 2 B m − n + j ( a I ) = j − 2 ( m − n + j )! 0 otherwise j = n − m | I | = j where the inner sum is over all subsets I ⊆ { 1 , 2 , . . . , n } of cardinality j and a I := ( a i : i ∈ I ) . Corollary For n ≥ 3 and odd m ≥ n − 2 � � n + j − 4 � � n � n � m ! 3 if n = m = 3 B ( j ) m − n + j = j − 2 ( m − n + j )! j 0 otherwise j =2 Relations for Barnes Zeta Functions Matthias Beck 4

Bernoulli–Barnes Relations � z n B k ( a ) z k a = ( a 1 , a 2 , . . . , a n ) ∈ R n ( e a 1 z − 1) · · · ( e a n z − 1) = k ! , > 0 k ≥ 0 Theorem 1 For n ≥ 3 and odd m ≥ 1 � � n + j − 4 � n � � 1 1 if n = m = 3 2 B m − n + j ( a I ) = j − 2 ( m − n + j )! 0 otherwise j = n − m | I | = j where the inner sum is over all subsets I ⊆ { 1 , 2 , . . . , n } of cardinality j , and a I := ( a i : i ∈ I ) . Poof Don’t use a Siegel-type integration path with integrand z s − 1 ( e a 1 z − 1) ( e a 2 z − 1) · · · ( e a n z − 1) Relations for Barnes Zeta Functions Matthias Beck 4

Bernoulli–Barnes Relations � z n B k ( a ) z k a = ( a 1 , a 2 , . . . , a n ) ∈ R n ( e a 1 z − 1) · · · ( e a n z − 1) = k ! , > 0 k ≥ 0 Theorem 1 For n ≥ 3 and odd m ≥ 1 � � n + j − 4 � n � � 1 1 if n = m = 3 2 B m − n + j ( a I ) = j − 2 ( m − n + j )! 0 otherwise j = n − m | I | = j where the inner sum is over all subsets I ⊆ { 1 , 2 , . . . , n } of cardinality j , and a I := ( a i : i ∈ I ) . Proof idea Show that � n + j − 4 � n � ( − z ) n − j � z | I | e z � i ∈ I a i � i ∈ I ( e a i z − 1) j − 2 j =2 | I | = j is an even function of z . Relations for Barnes Zeta Functions Matthias Beck 4

Barnes Zeta Functions � 1 ζ n ( z, x ; a ) := ( x + m 1 a 1 + · · · + m n a n ) z m ∈ Z n ≥ 0 defined for Re( x ) > 0 , Re( z ) > n and continued meromorphically to C . a = (1 , 1 , . . . , 1) − → ζ n ( s ; x ) := ζ ( s ; x, (1 , . . . , 1)) is the Hurwitz zeta function of order n . The Hurwitz zeta function is the special case n = 1 , the Riemmann zeta function the special case x = 1 . Relations for Barnes Zeta Functions Matthias Beck 5

Barnes Zeta Functions � 1 ζ n ( z, x ; a ) := ( x + m 1 a 1 + · · · + m n a n ) z m ∈ Z n ≥ 0 defined for Re( x ) > 0 , Re( z ) > n and continued meromorphically to C . a = (1 , 1 , . . . , 1) − → ζ n ( s ; x ) := ζ ( s ; x, (1 , . . . , 1)) is the Hurwitz zeta function of order n . The Hurwitz zeta function is the special case n = 1 , the Riemmann zeta function the special case x = 1 . ζ n ( − k, x ; a ) = ( − 1) n k ! ( k + n )! B k + n ( x ; a ) where B k ( x ; a ) is a Bernoulli–Barnes polynomial defined through � z n e xz B k ( x ; a ) z k ( e a 1 z − 1) · · · ( e a n z − 1) = k ! k ≥ 0 Note that B k ( a ) = B k (0; a ) Relations for Barnes Zeta Functions Matthias Beck 5

Barnes Zeta Relations � 1 ζ n ( z, x ; a ) := ( x + m 1 a 1 + · · · + m n a n ) z m ∈ Z n ≥ 0 � z n e xz B k ( x ; a ) z k ( e a 1 z − 1) · · · ( e a n z − 1) = k ! k ≥ 0 Theorem 2 Let a 1 , . . . , a n be pairwise coprime positive integers. Then � n − 1 � n − 1 � ( − 1) n − 1 ( − 1) k ζ ( s ; x, a ) = B n − 1 − k ( x ; a ) ζ ( s − k ; x ) ( n − 1)! k k =0 � � a j − 1 n � � s ; x + r a − s + σ − r ( a 1 , . . . , � a j , . . . , a n ; a j ) ζ j a j j =1 r =0 a j − 1 � e 2 πimr/a j a j . . . , a n ; a j ) := 1 � 1 − e 2 πima k /a j � where σ r ( a 1 , . . . , � � a j k � = j m =1 is a Fourier–Dedekind sum. Relations for Barnes Zeta Functions Matthias Beck 6

Reciprocity Theorems Theorem 2 Let a 1 , . . . , a n be pairwise coprime positive integers. Then � n − 1 � n − 1 � ( − 1) n − 1 ( − 1) k ζ ( s ; x, a ) = B n − 1 − k ( x ; a ) ζ ( s − k ; x ) ( n − 1)! k k =0 � � a j − 1 n � � s ; x + r a − s + σ − r ( a 1 , . . . , � a j , . . . , a n ; a j ) ζ . j a j j =1 r =0 Corollary [ n = 2 ] Let a, b be coprime positive integers. Then � � ζ ( s ; x, ( a, b )) = 1 1 − x abζ ( s − 1; x ) + ζ ( s ; x ) ab � b − 1 r � � � � a − 1 r � � � a − 1 a − 1 � � s ; x + r s ; x + r − a − s − b − s ζ ζ . a a b b r =0 r =0 Relations for Barnes Zeta Functions Matthias Beck 7

Reciprocity Theorems Corollary [ n = 2 ] Let a, b be coprime positive integers. Then � � ζ ( s ; x, ( a, b )) = 1 1 − x abζ ( s − 1; x ) + ζ ( s ; x ) ab � b − 1 r � � � � a − 1 r � � � a − 1 a − 1 � � s ; x + r s ; x + r − a − s − b − s ζ ζ . a a b b r =0 r =0 Corollary [ s ∈ Z < 0 ] Let a, b be coprime positive integers. Then � b − 1 r � � x + r � � a − 1 r � � x + r � a − 1 a − 1 � � a m + b m = B m +1 B m +1 a a b b r =0 r =0 � x � m + 2 B m +2 ( x, ( a, b )) + 1 1 m + 1 m + 2 B m +2 ( x ) + ab − 1 B m +1 ( x ) . ab This is reminiscent of reciprocity theorems for Dedekind sums. . . Relations for Barnes Zeta Functions Matthias Beck 7

Reciprocity Theorems � b − 1 r � � x + r � � a − 1 r � � x + r � a − 1 a − 1 � � a m + b m = B m +1 B m +1 a a b b r =0 r =0 � x � m + 2 B m +2 ( x, ( a, b )) + 1 1 m + 1 m + 2 B m +2 ( x ) + ab − 1 B m +1 ( x ) ab is a polynomial generalization of Apostol’s reciprocity law � � 1 a m − 1 s m ( a, b ) + b m − 1 s m ( b, a ) = m � m + 1 � m +1 � ( − 1) m +1 − i a i b m +1 − i B i B m +1 − i i i =0 for � a − 1 r � � r � �� ar �� a − 1 a − 1 � � r S m ( a, b ) := = B m b B m . b b b r =0 r =0 The case m = 1 gives Dedekind sums and their reciprocity law. Relations for Barnes Zeta Functions Matthias Beck 8

Hurwitz Zeta Relations Theorem 2 Let a 1 , . . . , a n be pairwise coprime positive integers. Then � n − 1 � n − 1 � ( − 1) n − 1 ( − 1) k ζ ( s ; x, a ) = B n − 1 − k ( x ; a ) ζ ( s − k ; x ) ( n − 1)! k k =0 � � a j − 1 n � � s ; x + r a − s + σ − r ( a 1 , . . . , � a j , . . . , a n ; a j ) ζ . j a j j =1 r =0 Corollary [ a = (1 , 1 , . . . , 1) ] � n − 1 � n − 1 � ζ n ( s ; x ) = ( − 1) n − 1 B ( n ) ( − 1) k n − 1 − k ( x ) ζ ( s − k ; x ) ( n − 1)! k k =0 Relations for Barnes Zeta Functions Matthias Beck 9