The non-vanishing spectrum of arithmetic progressions of squares Thomas A. Hulse Boston College Joint with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker 26 September, 2020 Universit´ e Laval Qu´ ebec-Maine Number Theory Conference 2020
Arithmetic Progressions of Squares
Consider a primitive, length-three arithmetic progression of square integers , { a 2 , b 2 , c 2 } , 1
Consider a primitive, length-three arithmetic progression of square integers , { a 2 , b 2 , c 2 } , that is: b 2 − a 2 = c 2 − b 2 with a , b , c ∈ N , a ≤ b ≤ c and all three integers are pairwise relatively prime. 1
Consider a primitive, length-three arithmetic progression of square integers , { a 2 , b 2 , c 2 } , that is: b 2 − a 2 = c 2 − b 2 with a , b , c ∈ N , a ≤ b ≤ c and all three integers are pairwise relatively prime. We observe that length-three is the shortest length sequence for which the term arithmetic progression is at all meaningful. 1
Consider a primitive, length-three arithmetic progression of square integers , { a 2 , b 2 , c 2 } , that is: b 2 − a 2 = c 2 − b 2 with a , b , c ∈ N , a ≤ b ≤ c and all three integers are pairwise relatively prime. We observe that length-three is the shortest length sequence for which the term arithmetic progression is at all meaningful. We also know (due to Euler [1] , among others) that there are not any such nontrivial progressions longer than length-three. So henceforth we are going to just drop the term length-three as being redundant, and we will further abbreviate arithmetic progressions as APs. 1
Consider a primitive, length-three arithmetic progression of square integers , { a 2 , b 2 , c 2 } , that is: b 2 − a 2 = c 2 − b 2 with a , b , c ∈ N , a ≤ b ≤ c and all three integers are pairwise relatively prime. We observe that length-three is the shortest length sequence for which the term arithmetic progression is at all meaningful. We also know (due to Euler [1] , among others) that there are not any such nontrivial progressions longer than length-three. So henceforth we are going to just drop the term length-three as being redundant, and we will further abbreviate arithmetic progressions as APs. We have devised a way of counting the number of APs of primitive integer squares given certain restrictions to the size of the integers. 1
Since b 2 − a 2 = c 2 − b 2 , 2
Since b 2 − a 2 = c 2 − b 2 , we observe that � a � 2 � c � 2 a 2 + c 2 = 2 b 2 ⇔ + = 2 . b b So we have a bijection between primitive APs of integer squares and √ rational points in an octant of a circle of radius 2. 2
Since b 2 − a 2 = c 2 − b 2 , we observe that � a � 2 � c � 2 a 2 + c 2 = 2 b 2 ⇔ + = 2 . b b So we have a bijection between primitive APs of integer squares and √ rational points in an octant of a circle of radius 2. 2
With this correspondence in mind, we state our first theorem, 3
With this correspondence in mind, we state our first theorem, Theorem 1 (H., Kuan, Lowry-Duda, Walker, 2020) [2] Fix δ ∈ [0 , 1]. For any ǫ > 0, the number of primitive APs of squares { a 2 , b 2 , c 2 } with b 2 ≤ X and ( a / b ) 2 ≤ δ is � 2 1 3 2 + O ǫ ( X 8 + ǫ ) . π 2 arcsin( δ/ 2) X 3
With this correspondence in mind, we state our first theorem, Theorem 1 (H., Kuan, Lowry-Duda, Walker, 2020) [2] Fix δ ∈ [0 , 1]. For any ǫ > 0, the number of primitive APs of squares { a 2 , b 2 , c 2 } with b 2 ≤ X and ( a / b ) 2 ≤ δ is � 2 1 3 2 + O ǫ ( X 8 + ǫ ) . π 2 arcsin( δ/ 2) X and observe it can also be stated as an equidistribution result: Theorem 1 (again) � a � b , c For any ǫ > 0, the number of reduced rational points on a circle √ b with radius 2 with b ≤ X within a sector of angle ω is 2 ω 3 4 + ǫ ) . π 2 X + O ǫ ( X 3
With this correspondence in mind, we state our first theorem, Theorem 1 (H., Kuan, Lowry-Duda, Walker, 2020) [2] Fix δ ∈ [0 , 1]. For any ǫ > 0, the number of primitive APs of squares { a 2 , b 2 , c 2 } with b 2 ≤ X and ( a / b ) 2 ≤ δ is � 2 1 3 2 + O ǫ ( X 8 + ǫ ) . π 2 arcsin( δ/ 2) X and observe it can also be stated as an equidistribution result: Theorem 1 (again) � a � b , c For any ǫ > 0, the number of reduced rational points on a circle √ b with radius 2 with b ≤ X within a sector of angle ω is 2 ω 3 4 + ǫ ) . π 2 X + O ǫ ( X The main term of this asymptotic is not difficult to see using elementary methods [5] , but the error term is nontrivial. 3
Let “APs” mean primitive arithmetic progressions of squares { a 2 , b 2 , c 2 } . 4
Let “APs” mean primitive arithmetic progressions of squares { a 2 , b 2 , c 2 } . Theorem 2 (H., Kuan, Lowry-Duda, Walker, 2020) [2] The number of APs with c 2 ≤ X is √ √ � 8 + ǫ � 2 1 3 2 + O ǫ π 2 log(1 + 2) X X . 4
Let “APs” mean primitive arithmetic progressions of squares { a 2 , b 2 , c 2 } . Theorem 2 (H., Kuan, Lowry-Duda, Walker, 2020) [2] The number of APs with c 2 ≤ X is √ √ � 8 + ǫ � 2 1 3 2 + O ǫ π 2 log(1 + 2) X X . Theorem 3 (H., Kuan, Lowry-Duda, Walker, 2020) [2] Suppose that Y ≤ X . The number of APs with a 2 ≤ Y and b 2 ≤ X is √ √ � � � 8 + ǫ � 1 2 log( e (4 − 2 2)) 1 1 3 2 log 2 + O ǫ √ 2 π 2 Y X / Y + Y X ǫ Y . π 2 4
Let “APs” mean primitive arithmetic progressions of squares { a 2 , b 2 , c 2 } . Theorem 2 (H., Kuan, Lowry-Duda, Walker, 2020) [2] The number of APs with c 2 ≤ X is √ √ � 8 + ǫ � 2 1 3 2 + O ǫ π 2 log(1 + 2) X X . Theorem 3 (H., Kuan, Lowry-Duda, Walker, 2020) [2] Suppose that Y ≤ X . The number of APs with a 2 ≤ Y and b 2 ≤ X is √ √ � � � 8 + ǫ � 1 2 log( e (4 − 2 2)) 1 1 3 2 log 2 + O ǫ √ 2 π 2 Y X / Y + Y X ǫ Y . π 2 Theorem 4 (H., Kuan, Lowry-Duda, Walker, 2020) [2] The number of primitive APs with ab ≤ X is √ � 8 + ǫ � 2 2 1 3 2 F 1 ( 1 4 , 1 2 , 5 4 , 1 2 + O ǫ 2 ) X X . π 2 4
Each of the previous asymptotic results are obtained via careful study of the shifted multiple Dirichlet series, 5
Each of the previous asymptotic results are obtained via careful study of the shifted multiple Dirichlet series, ∞ � r 1 ( h ) r 1 ( m ) r 1 (2 m − h ) D ( s , w ) := m s h w m , n =1 ( m , n )=1 where r 1 ( n ) is the number of ways n can be written as the square of an integer. So the coefficients of each summand essentially determine whether or not { h , m , 2 m − h } is an arithmetic progression of primitive squares since m − h = (2 m − h ) − m . 5
Each of the previous asymptotic results are obtained via careful study of the shifted multiple Dirichlet series, ∞ � r 1 ( h ) r 1 ( m ) r 1 (2 m − h ) D ( s , w ) := m s h w m , n =1 ( m , n )=1 where r 1 ( n ) is the number of ways n can be written as the square of an integer. So the coefficients of each summand essentially determine whether or not { h , m , 2 m − h } is an arithmetic progression of primitive squares since m − h = (2 m − h ) − m . In particular, we are able to derive a meromorphic continuation of D ( s , w ) to all ( s , w ) ∈ C 2 by means of a spectral expansion. Once we have a thorough understanding of the analytic behavior of the above series, we can obtain our aforementioned asymptotic results by carefully taking inverse Mellin transforms. To do this, we will take advantage of the automorphic properties of theta functions. 5
Theta Functions
Theta Functions Let H ⊂ C denote the upper-half plane , H := { z ∈ C | ℑ ( z ) > 0 } . 6
Theta Functions Let H ⊂ C denote the upper-half plane , H := { z ∈ C | ℑ ( z ) > 0 } . For N ∈ N , let Γ 0 ( N ) denote the congruence subgroup: �� � � � � A B � Γ 0 ( N ) := ∈ SL 2 ( Z ) � N | C . � C D 6
Theta Functions Let H ⊂ C denote the upper-half plane , H := { z ∈ C | ℑ ( z ) > 0 } . For N ∈ N , let Γ 0 ( N ) denote the congruence subgroup: �� � � � � A B � Γ 0 ( N ) := ∈ SL 2 ( Z ) � N | C . � C D It is easy to show that Γ 0 ( N ) acts on H by M¨ obius Maps: � � z = Az + B A B Cz + D . C D 6
Suppose for z ∈ H we define the theta function : 7
Suppose for z ∈ H we define the theta function : ∞ ∞ � � � e 2 π in 2 z = r 1 ( n ) e 2 π inz = 1 + r 1 ( n ) e 2 π inz θ ( z ) := n ∈ Z n =0 n =1 which is uniformly convergent on compact subsets of H . 7
Suppose for z ∈ H we define the theta function : ∞ ∞ � � � e 2 π in 2 z = r 1 ( n ) e 2 π inz = 1 + r 1 ( n ) e 2 π inz θ ( z ) := n ∈ Z n =0 n =1 which is uniformly convergent on compact subsets of H . � A B � For γ = ∈ Γ 0 (4), applying Poisson’s summation formula on the C D generators of Γ 0 (4) allows us to prove that � C � √ ǫ − 1 θ ( γ z ) = Cz + D θ ( z ) , D D � C � where denotes Shimura’s extension of the Jacobi symbol and ǫ D = 1 D or i depending on if D ≡ 1 or 3 (mod 4), respectively. [4] 7
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