Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: - PowerPoint PPT Presentation

Honors Combinatorics CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 7, Tuesday, May 19, 2020 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ 1 � n 2 n = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ ∞ 1 1 � � n 2 < 1 + n ( n − 1 ) n = 1 n = 2 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ ∞ ∞ 1 1 n − 1 − 1 1 � � � � � n 2 < 1 + n ( n − 1 ) = 1 + = 2 n n = 1 n = 2 n = 2 Theorem (Euler, 1734) ∞ n 2 = π 2 1 � 6 n = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ ∞ ∞ 1 1 n − 1 − 1 1 � � � � � n 2 < 1 + n ( n − 1 ) = 1 + = 2 n n = 1 n = 2 n = 2 Theorem (Euler, 1734) ∞ n 2 = π 2 1 � 6 ≈ 1 . 645 n = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ n 2 = π 2 1 � 6 n = 1 Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 n = 1 1 / n 2 ≥ π 2 / 6 Lemma 1 = ⇒ lower bound � ∞ Proof. 0 < α < π/ 2 = ⇒ α < tan α CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ n 2 = π 2 1 � 6 n = 1 Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 n = 1 1 / n 2 ≥ π 2 / 6 Lemma 1 = ⇒ lower bound � ∞ Proof. 0 < α < π/ 2 = ⇒ α < tan α m ( 2 m + 1 ) 2 > m ( 2 m − 1 ) � Lemma = ⇒ ∴ π 2 k 2 3 k = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares ∞ n 2 = π 2 1 � 6 n = 1 Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 n = 1 1 / n 2 ≥ π 2 / 6 Lemma 1 = ⇒ lower bound � ∞ Proof. 0 < α < π/ 2 = ⇒ α < tan α m ( 2 m + 1 ) 2 > m ( 2 m − 1 ) � Lemma = ⇒ ∴ π 2 k 2 3 k = 1 m k 2 > π 2 ( 2 m + 1 ) 2 → π 2 1 3 · m ( 2 m − 1 ) � 6 k = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 2 m = 2 m ( m + 1 ) 1 � sin 2 3 k π k = 1 2 m + 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 2 m = 2 m ( m + 1 ) 1 � sin 2 3 k π k = 1 2 m + 1 n = 1 1 / n 2 ≤ π 2 / 6 Lemma 2 = ⇒ upper bound � ∞ CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. De Moivre formula ( cos α + i sin α ) n = cos ( n α ) + i sin ( n α ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. De Moivre formula ( cos α + i sin α ) n = cos ( n α ) + i sin ( n α ) � � � � � � n n n sin 3 α cos n − 3 α + sin 5 α cos n − 5 sin α cos n − 1 α − sin ( n α ) = 1 3 5 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. De Moivre formula ( cos α + i sin α ) n = cos ( n α ) + i sin ( n α ) � � � � � � n n n sin 3 α cos n − 3 α + sin 5 α cos n − 5 sin α cos n − 1 α − sin ( n α ) = 1 3 5 cot α = cos α sin α = ⇒ CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. De Moivre formula ( cos α + i sin α ) n = cos ( n α ) + i sin ( n α ) � � � � � � n n n sin 3 α cos n − 3 α + sin 5 α cos n − 5 sin α cos n − 1 α − sin ( n α ) = 1 3 5 cot α = cos α sin α = ⇒ �� � � � � � � n n n sin ( n α ) = sin n α cot n − 1 α − cot n − 3 α + cot n − 5 α − . . . 1 3 5 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. �� � � � � � � n n n sin ( n α ) = sin n α cot n − 1 α − cot n − 3 α + cot n − 5 α − . . . 1 3 5 n := 2 m + 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. �� � � � � � � n n n sin ( n α ) = sin n α cot n − 1 α − cot n − 3 α + cot n − 5 α − . . . 1 3 5 n := 2 m + 1 � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − 1 + x m − x m − 2 − . . . P ( x ) := 1 3 5 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. �� � � � � � � n n n sin ( n α ) = sin n α cot n − 1 α − cot n − 3 α + cot n − 5 α − . . . 1 3 5 n := 2 m + 1 � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − 1 + x m − x m − 2 − . . . P ( x ) := 1 3 5 sin (( 2 m + 1 ) α ) = sin 2 m + 1 α · P ( cot 2 α ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 sin (( 2 m + 1 ) α ) = sin 2 m + 1 α · P ( cot 2 α ) Roots of P ? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 sin (( 2 m + 1 ) α ) = sin 2 m + 1 α · P ( cot 2 α ) k π Roots of P ? LHS vanishes at 2 m + 1 k π ∴ roots of P are r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares Lemma 1 m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 Proof. � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 sin (( 2 m + 1 ) α ) = sin 2 m + 1 α · P ( cot 2 α ) k π Roots of P ? LHS vanishes at 2 m + 1 k π ∴ roots of P are r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 these are all the roots P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) k π r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) k π r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 ( 2 m + 1 )( r 1 + · · · + r m ) = ( 2 m + 1 Coeff ( x m − 1 ) 3 ) , i.e., CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) k π r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 ( 2 m + 1 )( r 1 + · · · + r m ) = ( 2 m + 1 Coeff ( x m − 1 ) 3 ) , i.e., m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 3 k = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) k π r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 ( 2 m + 1 )( r 1 + · · · + r m ) = ( 2 m + 1 Coeff ( x m − 1 ) 3 ) , i.e., m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 QED[Lemma 1] 3 k = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Sum of inverse squares � � � � � � 2 m + 1 2 m + 1 2 m + 1 x m − x m − 1 + x m − 2 − . . . P ( x ) := 1 3 5 P ( x ) = ( 2 m + 1 )( x − r 1 ) · · · ( x − r m ) k π r k = cot 2 ( k = 1 , . . . , m ) 2 m + 1 ( 2 m + 1 )( r 1 + · · · + r m ) = ( 2 m + 1 Coeff ( x m − 1 ) 3 ) , i.e., m 2 m + 1 = m ( 2 m − 1 ) k π � cot 2 QED[Lemma 1] 3 k = 1 Matoušek – Nešetˇ ril: Source: Invitation to Discrete Mathematics CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Generating functions: Number partitions 5 = 5 5 = 4 + 1 5 = 3 + 2 5 = 3 + 1 + 1 5 = 2 + 2 + 1 5 = 2 + 1 + 1 + 1 5 = 1 + 1 + 1 + 1 + 1 p ( n ) = number of partitions of the number n CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Generating functions: Number partitions 5 = 5 5 = 4 + 1 5 = 3 + 2 5 = 3 + 1 + 1 5 = 2 + 2 + 1 5 = 2 + 1 + 1 + 1 5 = 1 + 1 + 1 + 1 + 1 p ( n ) = number of partitions of the number n E.g., p ( 5 ) = 7 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Generating functions: Number partitions 5 = 5 5 = 4 + 1 5 = 3 + 2 5 = 3 + 1 + 1 5 = 2 + 2 + 1 5 = 2 + 1 + 1 + 1 5 = 1 + 1 + 1 + 1 + 1 p ( n ) = number of partitions of the number n E.g., p ( 5 ) = 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 5 11 15 22 30 42 56 77 101 135 7 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Generating functions: Number partitions Theorem (Hardy–Ramanujan 1917) 1 √ 2 n / 3 e π p ( n ) ∼ √ 4 3 n CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics