Happy 103rd birthday, Richard Guy Karl Dilcher Infinite products - PowerPoint PPT Presentation

Happy 103rd birthday, Richard Guy Karl Dilcher Infinite products

Infinite products involving Dirichlet characters and cyclotomic polynomials Karl Dilcher Dalhousie Number Theory Seminar, Sept. 30, 2019 Karl Dilcher Infinite products

Joint work with Christophe Vignat (Université d’Orsay and Tulane University) Karl Dilcher Infinite products

1. Introduction Well-known fact about infinite products: ∞ � 1 − 1 � � k k = 2 diverges. Karl Dilcher Infinite products

1. Introduction Well-known fact about infinite products: ∞ � 1 − 1 � � k k = 2 diverges. On the other hand, e.g., ∞ √ 1 − ( − 1 ) k � � = π � 2 . 2 k + 1 4 k = 1 Karl Dilcher Infinite products

1. Introduction Well-known fact about infinite products: ∞ � 1 − 1 � � k k = 2 diverges. On the other hand, e.g., ∞ √ 1 − ( − 1 ) k � � = π � 2 . 2 k + 1 4 k = 1 Related: Weierstrass factorization theorem which gives, e.g., ∞ 1 1 + z � � e − z / k = ze γ z � Γ( z ) . k k = 1 Karl Dilcher Infinite products

1. Introduction Well-known fact about infinite products: ∞ � 1 − 1 � � k k = 2 diverges. On the other hand, e.g., ∞ √ 1 − ( − 1 ) k � � = π � 2 . 2 k + 1 4 k = 1 Related: Weierstrass factorization theorem which gives, e.g., ∞ 1 1 + z � � e − z / k = ze γ z � Γ( z ) . k k = 1 First indication that the gamma function may be involved. Karl Dilcher Infinite products

Source: Wikipedia, “Gamma Function". Karl Dilcher Infinite products

Source: Wikipedia, “Gamma Function". Karl Dilcher Infinite products

Karl Dilcher Infinite products

A general result: A convergent infinite product of a rational function in the index k can always be written as a product or quotient of finitely many values of the gamma function. The last identity is an ingredient in the proof of this fact. Karl Dilcher Infinite products

A general result: A convergent infinite product of a rational function in the index k can always be written as a product or quotient of finitely many values of the gamma function. The last identity is an ingredient in the proof of this fact. Goal of this talk: To extend the identity ∞ √ 1 − ( − 1 ) k � � = π � 2 2 k + 1 4 k = 1 in a different direction. Karl Dilcher Infinite products

2. Main Result Let χ be the unique nontrivial Dirichlet character modulo 4, i.e., the periodic function of period 4 defined by χ ( 1 ) = 1 , χ ( 3 ) = − 1, and χ ( 0 ) = χ ( 2 ) = 0. Karl Dilcher Infinite products

2. Main Result Let χ be the unique nontrivial Dirichlet character modulo 4, i.e., the periodic function of period 4 defined by χ ( 1 ) = 1 , χ ( 3 ) = − 1, and χ ( 0 ) = χ ( 2 ) = 0. Then we can rewrite this last identity as ∞ √ � 1 − χ ( k ) � = π � 2 , (1) k 4 k = 2 Karl Dilcher Infinite products

2. Main Result Let χ be the unique nontrivial Dirichlet character modulo 4, i.e., the periodic function of period 4 defined by χ ( 1 ) = 1 , χ ( 3 ) = − 1, and χ ( 0 ) = χ ( 2 ) = 0. Then we can rewrite this last identity as ∞ √ � 1 − χ ( k ) � = π � 2 , (1) k 4 k = 2 and as a function of the complex variable z as √ ∞ 1 − χ ( k ) z 1 − z sin ( 1 − z ) π 2 � � � = . (2) k 4 k = 2 Karl Dilcher Infinite products

2. Main Result Let χ be the unique nontrivial Dirichlet character modulo 4, i.e., the periodic function of period 4 defined by χ ( 1 ) = 1 , χ ( 3 ) = − 1, and χ ( 0 ) = χ ( 2 ) = 0. Then we can rewrite this last identity as ∞ √ � 1 − χ ( k ) � = π � 2 , (1) k 4 k = 2 and as a function of the complex variable z as √ ∞ 1 − χ ( k ) z 1 − z sin ( 1 − z ) π 2 � � � = . (2) k 4 k = 2 Note: (2) implies (1) by letting z → 1. Karl Dilcher Infinite products

2. Main Result Let χ be the unique nontrivial Dirichlet character modulo 4, i.e., the periodic function of period 4 defined by χ ( 1 ) = 1 , χ ( 3 ) = − 1, and χ ( 0 ) = χ ( 2 ) = 0. Then we can rewrite this last identity as ∞ √ � 1 − χ ( k ) � = π � 2 , (1) k 4 k = 2 and as a function of the complex variable z as √ ∞ 1 − χ ( k ) z 1 − z sin ( 1 − z ) π 2 � � � = . (2) k 4 k = 2 Note: (2) implies (1) by letting z → 1. This is a special case of the following result. Karl Dilcher Infinite products

Theorem 1 Let χ be a primitive nonprincipal Dirichlet character with conductor q > 2 . Then q − 1 ∞ = ( 2 π ) ϕ ( q ) / 2 1 − χ ( k ) z 1 � � � � ( 1 − z ) ǫ ( q ) · � , � k j − χ ( j ) z Γ k = 2 j = 1 q ( j , q )= 1 Karl Dilcher Infinite products

Theorem 1 Let χ be a primitive nonprincipal Dirichlet character with conductor q > 2 . Then q − 1 ∞ = ( 2 π ) ϕ ( q ) / 2 1 − χ ( k ) z 1 � � � � ( 1 − z ) ǫ ( q ) · � , � k j − χ ( j ) z Γ k = 2 j = 1 q ( j , q )= 1 where ǫ ( q ) is defined by � √ p when q is a power of a prime p , ǫ ( q ) = otherwise . 1 Karl Dilcher Infinite products

Main ingredients in proof: 1. Infinite product expansion for 1 / Γ( z ) leads to ∞ Γ( u ) � v � Γ( u + v ) = e γ v � e − v / ( k + 1 ) , 1 + u + k k = 0 Karl Dilcher Infinite products

Main ingredients in proof: 1. Infinite product expansion for 1 / Γ( z ) leads to ∞ Γ( u ) � v � Γ( u + v ) = e γ v � e − v / ( k + 1 ) , 1 + u + k k = 0 and this, in turn, gives rise to Lemma 2 Let n ∈ N , a , z 1 , . . . , z n ∈ C with z j � = 0 for j = 1 , . . . , n, and let f : { 1 , 2 , . . . , n } → C satisfy f ( 1 ) + · · · + f ( n ) = 0 . Then Karl Dilcher Infinite products

Main ingredients in proof: 1. Infinite product expansion for 1 / Γ( z ) leads to ∞ Γ( u ) � v � Γ( u + v ) = e γ v � e − v / ( k + 1 ) , 1 + u + k k = 0 and this, in turn, gives rise to Lemma 2 Let n ∈ N , a , z 1 , . . . , z n ∈ C with z j � = 0 for j = 1 , . . . , n, and let f : { 1 , 2 , . . . , n } → C satisfy f ( 1 ) + · · · + f ( n ) = 0 . Then ∞ n n � a � Γ( z j ) � � � 1 − f ( j ) = Γ( z j − f ( j ) a ) . z j + k k = 0 j = 1 j = 1 Karl Dilcher Infinite products

2. Products of certain gamma function values: Karl Dilcher Infinite products

2. Products of certain gamma function values: Lemma 3 (Chamberland and Straub, 2013) For any integer n ≥ 2 and prime p we have � j n − 1 � ( 2 π ) ϕ ( n ) / 2 if n is not a prime power , � � Γ = 1 √ p ( 2 π ) ϕ ( n ) / 2 if n = p ν , ν ≥ 1 . n j = 1 ( j , n )= 1 Karl Dilcher Infinite products

2. Products of certain gamma function values: Lemma 3 (Chamberland and Straub, 2013) For any integer n ≥ 2 and prime p we have � j n − 1 � ( 2 π ) ϕ ( n ) / 2 if n is not a prime power , � � Γ = 1 √ p ( 2 π ) ϕ ( n ) / 2 if n = p ν , ν ≥ 1 . n j = 1 ( j , n )= 1 This extends the well-known identity � j n − 1 = ( 2 π ) ( n − 1 ) / 2 � � Γ √ n . n j = 1 Karl Dilcher Infinite products

Marc (left) and Armin (middle) after being hit by a rogue wave in Peggy’s Cove, Nova Scotia. Karl Dilcher Infinite products

Using the reflection formula π Γ( z )Γ( 1 − z ) = sin ( π z ) , z � = 0 , ± 1 , ± 2 , . . . : Karl Dilcher Infinite products

Using the reflection formula π Γ( z )Γ( 1 − z ) = sin ( π z ) , z � = 0 , ± 1 , ± 2 , . . . : Corollary 4 Let χ be an odd primitive Dirichlet character with conductor q > 2 . Then ⌊ q − 1 2 ⌋ ∞ 2 ϕ ( q ) / 2 � � 1 − χ ( k ) z π j − χ ( j ) z � � � � = ( 1 − z ) ǫ ( q ) · sin , k q k = 2 j = 1 ( j , q )= 1 Karl Dilcher Infinite products

Using the reflection formula π Γ( z )Γ( 1 − z ) = sin ( π z ) , z � = 0 , ± 1 , ± 2 , . . . : Corollary 4 Let χ be an odd primitive Dirichlet character with conductor q > 2 . Then ⌊ q − 1 2 ⌋ ∞ 2 ϕ ( q ) / 2 � � 1 − χ ( k ) z π j − χ ( j ) z � � � � = ( 1 − z ) ǫ ( q ) · sin , k q k = 2 j = 1 ( j , q )= 1 and in particular, ⌊ q − 1 2 ⌋ ∞ = π 2 ϕ ( q ) / 2 � 1 − χ ( k ) � � π j − χ ( j ) � � � · sin . k q ǫ ( q ) q k = 2 j = 2 ( j , q )= 1 Karl Dilcher Infinite products

Example 1. Let q = 3. Then the only nonprincipal character is given by χ ( 1 ) = 1 and χ ( 2 ) = − 1. Then ∞ 2 1 − χ ( k ) z � � � · sin ( π √ = 3 ( 1 − z )) , k ( 1 − z ) 3 k = 2 Karl Dilcher Infinite products

Example 1. Let q = 3. Then the only nonprincipal character is given by χ ( 1 ) = 1 and χ ( 2 ) = − 1. Then ∞ 2 1 − χ ( k ) z � � � · sin ( π √ = 3 ( 1 − z )) , k ( 1 − z ) 3 k = 2 and with z = 1 and z = 1 2 we get, respectively, ∞ ∞ � � � � 1 − χ ( k ) = 2 π 1 − χ ( k ) 2 � � √ , = √ . k 2 k 3 3 3 k = 2 k = 2 Karl Dilcher Infinite products

Example 2. q = 5 is the smallest conductor that has nonreal characters. We choose the one (of two) that is given by χ ( 1 ) = 1, χ ( 2 ) = i , χ ( 3 ) = − i and χ ( 4 ) = − 1. Then Karl Dilcher Infinite products