Gauss congruences ICOMAS 2018: Special Session on Analytic Number - PowerPoint PPT Presentation

Gauss congruences ICOMAS 2018: Special Session on Analytic Number Theory The University of Memphis Armin Straub May 8, 2018 University of South Alabama based on joint work with and Frits Beukers Marc Houben (Utrecht University) (Utrecht University) Gauss congruences Armin Straub 1 / 13

Introduction: Diagonals • Given a series 1 · · · x n d � a ( n 1 , . . . , n d ) x n 1 F ( x 1 , . . . , x d ) = d , n 1 ,...,n d � 0 its diagonal coefficients are the coefficients a ( n, . . . , n ) . The diagonal coefficients of EG ∞ 1 � ( x + y ) n 1 − x − y = n =0 � 2 n � are the central binomial coefficients . n Gauss congruences Armin Straub 2 / 13

Introduction: Diagonals • Given a series 1 · · · x n d � a ( n 1 , . . . , n d ) x n 1 F ( x 1 , . . . , x d ) = d , n 1 ,...,n d � 0 its diagonal coefficients are the coefficients a ( n, . . . , n ) . The diagonal coefficients of EG ∞ 1 � ( x + y ) n 1 − x − y = n =0 � 2 n � are the central binomial coefficients . n For comparison, their univariate generating function is ∞ � 2 n � 1 x n = � √ 1 − 4 x. n n =0 Gauss congruences Armin Straub 2 / 13

Introduction: Rational generating functions 2 − x EG L n +1 = L n + L n − 1 The Lucas numbers L n have GF 1 − x − x 2 . L 0 = 2 , L 1 = 1 • The sequences with rational GF are precisely the C -finite ones. Gauss congruences Armin Straub 3 / 13

Introduction: Rational generating functions 2 − x EG L n +1 = L n + L n − 1 The Lucas numbers L n have GF 1 − x − x 2 . L 0 = 2 , L 1 = 1 • The sequences with rational GF are precisely the C -finite ones. n EG 1 � n �� n + k � The Delannoy numbers have GF 1 − 6 x + x 2 . √ � D n = k k k =0 1 They are the diagonal of 1 − x − y − xy . • The sequences with algebraic GF are precisely the diagonals of 2 -variable rational functions. Gauss congruences Armin Straub 3 / 13

Introduction: Rational generating functions 2 − x EG L n +1 = L n + L n − 1 The Lucas numbers L n have GF 1 − x − x 2 . L 0 = 2 , L 1 = 1 • The sequences with rational GF are precisely the C -finite ones. n EG 1 � n �� n + k � The Delannoy numbers have GF 1 − 6 x + x 2 . √ � D n = k k k =0 1 They are the diagonal of 1 − x − y − xy . • The sequences with algebraic GF are precisely the diagonals of 2 -variable rational functions. The diagonal of a rational function is D -finite. THM Gessel, More generally, the diagonal of a D -finite function is D -finite. Zeilberger, Lipshitz F ∈ K [[ x 1 , . . . , x d ]] is D -finite if its partial derivatives span a finite-dimensional 1981–88 vector space over K ( x 1 , . . . , x d ) . Gauss congruences Armin Straub 3 / 13

Introduction: Franel numbers n EG � 3 � n � The Franel numbers are the diagonal of k k =0 1 1 − x − y − z + 4 xyz . Their GF is � 1 3 , 2 27 x 2 � 1 � � 3 1 − 2 x 2 F 1 . � (1 − 2 x ) 3 1 � Gauss congruences Armin Straub 4 / 13

Introduction: Franel numbers n EG � 3 � n � The Franel numbers are the diagonal of k k =0 1 1 − x − y − z + 4 xyz . Their GF is � 1 3 , 2 27 x 2 � 1 � � 3 1 − 2 x 2 F 1 . � (1 − 2 x ) 3 1 � • Not at all unique! The Franel numbers are also the diagonal of 1 (1 − x )(1 − y )(1 − z ) − xyz . Gauss congruences Armin Straub 4 / 13

Introduction: Ap´ ery numbers as diagonals The Ap´ ery numbers are the diagonal coefficients of THM S 2014 1 . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 Gauss congruences Armin Straub 5 / 13

Introduction: Ap´ ery numbers as diagonals The Ap´ ery numbers are the diagonal coefficients of THM S 2014 1 . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 • Univariate generating function: � 1 2 , 1 2 , 1 17 − x − z � 1024 x � A ( n ) x n = � 2 � √ 2(1 + x + z ) 3 / 2 3 F 2 � − , � (1 − x + z ) 4 1 , 1 4 n � 0 √ 1 − 34 x + x 2 . where z = Gauss congruences Armin Straub 5 / 13

Introduction: Ap´ ery numbers as diagonals The Ap´ ery numbers are the diagonal coefficients of THM S 2014 1 . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 • Univariate generating function: � 1 2 , 1 2 , 1 17 − x − z � 1024 x � A ( n ) x n = � 2 � √ 2(1 + x + z ) 3 / 2 3 F 2 � − , � (1 − x + z ) 4 1 , 1 4 n � 0 √ 1 − 34 x + x 2 . where z = • Well-developed theory of multivariate asymptotics e.g., Pemantle–Wilson Gauss congruences Armin Straub 5 / 13

Introduction: Ap´ ery numbers as diagonals The Ap´ ery numbers are the diagonal coefficients of THM S 2014 1 . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 • Univariate generating function: � 1 2 , 1 2 , 1 17 − x − z � 1024 x � A ( n ) x n = � 2 � √ 2(1 + x + z ) 3 / 2 3 F 2 � − , � (1 − x + z ) 4 1 , 1 4 n � 0 √ 1 − 34 x + x 2 . where z = • Well-developed theory of multivariate asymptotics e.g., Pemantle–Wilson • Such diagonals are algebraic modulo p r . Furstenberg, Deligne ’67, ’84 Automatically leads to congruences such as � 1 (mod 8) , if n even , Chowla–Cowles–Cowles ’80 A ( n ) ≡ Rowland–Yassawi ’13 5 (mod 8) , if n odd . Gauss congruences Armin Straub 5 / 13

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. Gauss congruences Armin Straub 6 / 13

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . Gauss congruences Armin Straub 6 / 13

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . THM d ) a d ≡ 0 � µ ( m (mod m ) Gauss d | m obius function: µ ( n ) = ( − 1) # of p | n if n is square-free, µ ( n ) = 0 else M¨ Gauss congruences Armin Straub 6 / 13

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . THM d ) a d ≡ 0 � µ ( m (mod m ) Gauss d | m obius function: µ ( n ) = ( − 1) # of p | n if n is square-free, µ ( n ) = 0 else M¨ If m = p r then only d = p r , d = p r − 1 contribute, and we get EG a p r ≡ a p r − 1 (mod p r ) . Gauss congruences Armin Straub 6 / 13

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m Gauss congruences Armin Straub 7 / 13

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG Gauss congruences Armin Straub 7 / 13

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 Gauss congruences Armin Straub 7 / 13

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 n � n �� n + k � � • a ( n ) = D n Delannoy numbers : D n = k k k =0 Gauss congruences Armin Straub 7 / 13

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 n � n �� n + k � � • a ( n ) = D n Delannoy numbers : D n = k k k =0 • Later, we allow a ( n ) ∈ Q . If the Gauss congruences hold for all but finitely many p , we say that the sequence (or its GF) has the Gauss property . • Similarly, for multivariate sequences a ( n ) , we require a ( m p r ) ≡ a ( m p r − 1 ) (mod p r ) . Gauss congruences Armin Straub 7 / 13

More sequences satisfying Gauss congruences a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) (G) • realizable sequences a ( n ) , i.e., for some map T : X → X , a ( n ) = # { x ∈ X : T n x = x } “points of period n ” Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05 In fact, up to a positivity condition, (G) characterizes realizability. Gauss congruences Armin Straub 8 / 13

More sequences satisfying Gauss congruences a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) (G) • realizable sequences a ( n ) , i.e., for some map T : X → X , a ( n ) = # { x ∈ X : T n x = x } “points of period n ” Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05 In fact, up to a positivity condition, (G) characterizes realizability. • a ( n ) = trace( M n ) J¨ anichen ’21, Schur ’37; also: Arnold, Zarelua where M is an integer matrix Gauss congruences Armin Straub 8 / 13