Ramanujans Function Armin Straub 06-May 2007 Armin Straub - PowerPoint PPT Presentation

Introduction Modular Forms More Congruences for τ Fun Stuff Ramanujan’s τ Function Armin Straub 06-May 2007 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms More Congruences for τ Fun Stuff Outline Introduction The τ Function Simple Congruences for τ Modular Forms Eisenstein Series Differentiating Modular Forms More Congruences for τ Differentiating ∆ An Exact Formula Modulus 691 Modulus 7 Fun Stuff Ramanujan Can Err Open Problems Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff The τ Function (I) Definition (1 − q n ) 24 = � � τ ( n ) q n . ∆ � q n � 1 n � 1 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff The τ Function (I) Definition (1 − q n ) 24 = � � τ ( n ) q n . ∆ � q n � 1 n � 1 Example The first values are τ (1) = 1 τ (5) = 4830 τ (2) = − 24 τ (6) = − 6048 τ (3) = 252 τ (7) = − 16744 τ (4) = − 1472 τ (8) = 84480 . . . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff The τ Function (II) ◮ A plot of log ( τ ( n )) . 25 100 − 25 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff The τ Function (II) ◮ A plot of log ( τ ( n )) . 25 100 − 25 ◮ Ramanujan conjectured and Deligne proved | τ ( p ) | � 2 p 11 / 2 . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff The τ Function (II) ◮ A plot of log ( τ ( n )) . 25 100 − 25 ◮ Ramanujan conjectured and Deligne proved | τ ( p ) | � 2 p 11 / 2 . . . . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) ◮ Hence, � (1 − q n ) 24 � (1 − q n ) 3 � (1 − q 7 n ) 3 q ≡ q (mod 7) n � 1 n � 1 n � 1 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) ◮ Hence, using Jacobi’s identity � (1 − q n ) 24 � (1 − q n ) 3 � (1 − q 7 n ) 3 q ≡ q (mod 7) n � 1 n � 1 n � 1 � ( − 1) n (2 n + 1) q n ( n +1) / 2 � (1 − q 7 n ) 3 = q n � 0 n � 1 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) ◮ Hence, using Jacobi’s identity � (1 − q n ) 24 � (1 − q n ) 3 � (1 − q 7 n ) 3 q ≡ q (mod 7) n � 1 n � 1 n � 1 � ( − 1) n (2 n + 1) q n ( n +1) / 2 � (1 − q 7 n ) 3 = q n � 0 n � 1 ◮ n ( n +1) + 1 ≡ 0 , 1 , 2 , 4 (mod 7) 2 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) ◮ Hence, using Jacobi’s identity � (1 − q n ) 24 � (1 − q n ) 3 � (1 − q 7 n ) 3 q ≡ q (mod 7) n � 1 n � 1 n � 1 � ( − 1) n (2 n + 1) q n ( n +1) / 2 � (1 − q 7 n ) 3 = q n � 0 n � 1 ◮ n ( n +1) + 1 ≡ 0 , 1 , 2 , 4 (mod 7) 2 ◮ We conclude τ (7 n + 3) , τ (7 n + 5) , τ (7 n + 6) ≡ 0 (mod 7) . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (I) ◮ ( a + b ) p ≡ a p + b p (mod p ) ◮ Hence, using Jacobi’s identity � (1 − q n ) 24 � (1 − q n ) 3 � (1 − q 7 n ) 3 q ≡ q (mod 7) n � 1 n � 1 n � 1 � ( − 1) n (2 n + 1) q n ( n +1) / 2 � (1 − q 7 n ) 3 = q n � 0 n � 1 ◮ n ( n +1) + 1 ≡ 0 , 1 , 2 , 4 (mod 7) 2 ◮ We conclude τ (7 n ) , τ (7 n + 3) , τ (7 n + 5) , τ (7 n + 6) ≡ 0 (mod 7) . . . . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (II) Theorem τ ( mn ) = τ ( m ) τ ( n ) if gcd( m, n ) = 1 , τ ( p n +1 ) = τ ( p ) τ ( p n ) − p 11 τ ( p n − 1 ) if p prime . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (II) Theorem τ ( mn ) = τ ( m ) τ ( n ) if gcd( m, n ) = 1 , τ ( p n +1 ) = τ ( p ) τ ( p n ) − p 11 τ ( p n − 1 ) if p prime . ◮ τ ( p ) ≡ 0 (mod p ) = ⇒ τ ( np ) ≡ 0 (mod p ) Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (II) Theorem τ ( mn ) = τ ( m ) τ ( n ) if gcd( m, n ) = 1 , τ ( p n +1 ) = τ ( p ) τ ( p n ) − p 11 τ ( p n − 1 ) if p prime . ◮ τ ( p ) ≡ 0 (mod p ) = ⇒ τ ( np ) ≡ 0 (mod p ) ◮ τ (7) = − 16744 = ⇒ τ (7 n ) ≡ 0 (mod 7) Armin Straub Ramanujan’s τ Function

Introduction Modular Forms The τ Function More Congruences for τ Simple Congruences for τ Fun Stuff Simple Congruences (II) Theorem τ ( mn ) = τ ( m ) τ ( n ) if gcd( m, n ) = 1 , τ ( p n +1 ) = τ ( p ) τ ( p n ) − p 11 τ ( p n − 1 ) if p prime . ◮ τ ( p ) ≡ 0 (mod p ) = ⇒ τ ( np ) ≡ 0 (mod p ) ◮ τ (7) = − 16744 = ⇒ τ (7 n ) ≡ 0 (mod 7) ◮ Only a few such primes are known: 2 , 3 , 5 , 7 , 2411 . . . . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Eisenstein Series ◮ Eisenstein Series E n are an example of modular forms of weight 2 n . E n = 1 − 4 k � σ 2 k − 1 ( n ) q n B 2 k n � 1 Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Eisenstein Series ◮ Eisenstein Series E n are an example of modular forms of weight 2 n . E n = 1 − 4 k � σ 2 k − 1 ( n ) q n B 2 k n � 1 ◮ Any modular form f can be obtained as a polynomial in E 2 and E 3 . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Eisenstein Series ◮ Eisenstein Series E n are an example of modular forms of weight 2 n . E n = 1 − 4 k � σ 2 k − 1 ( n ) q n B 2 k n � 1 ◮ Any modular form f can be obtained as a polynomial in E 2 and E 3 . ◮ What about E 1 ? Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Eisenstein Series ◮ Eisenstein Series E n are an example of modular forms of weight 2 n . E n = 1 − 4 k � σ 2 k − 1 ( n ) q n B 2 k n � 1 ◮ Any modular form f can be obtained as a polynomial in E 2 and E 3 . ◮ What about E 1 ? ◮ E 1 is not modular but E 1 ( − 1 /z ) = z 2 E 1 ( z ) + 12 2 πiz. . . . Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Differentiating Modular Forms ◮ If f is a modular form. What about d f d z ? Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Differentiating Modular Forms ◮ If f is a modular form. What about q d f 1 d f d q = d z ? 2 πi Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Differentiating Modular Forms ◮ If f is a modular form. What about θf � q d f 1 d f d q = d z ? 2 πi Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Differentiating Modular Forms ◮ If f is a modular form. What about θf � q d f 1 d f d q = d z ? 2 πi ◮ Not modular, but this is where E 1 comes in. Armin Straub Ramanujan’s τ Function

Introduction Modular Forms Eisenstein Series More Congruences for τ Differentiating Modular Forms Fun Stuff Differentiating Modular Forms ◮ If f is a modular form. What about θf � q d f 1 d f d q = d z ? 2 πi ◮ Not modular, but this is where E 1 comes in. Lemma If f is a modular form of weight k then θf − k 12 E 1 f is a modular form of weight k + 2 . . . . Armin Straub Ramanujan’s τ Function

Introduction Differentiating ∆ Modular Forms An Exact Formula More Congruences for τ Modulus 691 Fun Stuff Modulus 7 θf − k Differentiating ∆ 12 E 1 f ◮ Applied to ∆ which is of weight 12 , θ ∆ − E 1 ∆ = 0 . Armin Straub Ramanujan’s τ Function