Arithmetic aspects of short random walks Number Theory Seminar - PowerPoint PPT Presentation

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 • On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 • On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 • On a supercomputer: Lawrence Berkeley National Laboratory, 256 cores W 5 (1) ≈ 2 . 0081618 Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � e 2 πix 1 + . . . + e 2 πix n � � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 • On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 • On a supercomputer: Lawrence Berkeley National Laboratory, 256 cores W 5 (1) ≈ 2 . 0081618 • Hard to evaluate numerically to high precision. Monte-Carlo integration gives approximations with an asymptotic error of √ O (1 / N ) where N is the number of sample points. Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 π Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Arithmetic aspects of short random walks Armin Straub 10 / 35

Moments of random walks The s th moment W n ( s ) of the density p n : DEF � ∞ � � s d x � � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = W n ( s ) := [0 , 1] n 0 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Arithmetic aspects of short random walks Armin Straub 10 / 35

Even moments n s = 0 s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 1 2 6 20 70 252 A000984 3 1 3 15 93 639 4653 A002893 4 1 4 28 256 2716 31504 A002895 5 1 5 45 545 7885 127905 A169714 6 1 6 66 996 18306 384156 A169715 Arithmetic aspects of short random walks Armin Straub 11 / 35

Even moments n s = 0 s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 1 2 6 20 70 252 A000984 3 1 3 15 93 639 4653 A002893 4 1 4 28 256 2716 31504 A002895 5 1 5 45 545 7885 127905 A169714 6 1 6 66 996 18306 384156 A169715 EG k � k � 2 � 2 j � � W 3 (2 k ) = j j j =0 k � k � 2 � 2 j �� 2( k − j ) � � W 4 (2 k ) = j j k − j j =0 j k � k � 2 � 2( k − j ) � � j � 2 � 2 ℓ � � � W 5 (2 k ) = j k − j ℓ ℓ j =0 ℓ =0 Arithmetic aspects of short random walks Armin Straub 11 / 35

A combinatorial formula for the even moments • s th moment W n ( s ) of the density p n : � � s d x � � e 2 πix 1 + . . . + e 2 πix n � W n ( s ) = [0 , 1] n � � 2 THM k � Borwein- W n (2 k ) = Nuyens- a 1 , . . . , a n S-Wan a 1 + ··· + a n = k 2010 Arithmetic aspects of short random walks Armin Straub 12 / 35

A combinatorial formula for the even moments • s th moment W n ( s ) of the density p n : � � s d x � � e 2 πix 1 + . . . + e 2 πix n � W n ( s ) = [0 , 1] n � � 2 THM k � Borwein- W n (2 k ) = Nuyens- a 1 , . . . , a n S-Wan a 1 + ··· + a n = k 2010 • W n (2 k ) counts the number of abelian squares : strings xy of length 2 k from an alphabet with n letters such that y is a permutation of x . Arithmetic aspects of short random walks Armin Straub 12 / 35

A combinatorial formula for the even moments • s th moment W n ( s ) of the density p n : � � s d x � � e 2 πix 1 + . . . + e 2 πix n � W n ( s ) = [0 , 1] n � � 2 THM k � Borwein- W n (2 k ) = Nuyens- a 1 , . . . , a n S-Wan a 1 + ··· + a n = k 2010 • W n (2 k ) counts the number of abelian squares : strings xy of length 2 k from an alphabet with n letters such that y is a permutation of x . • Introduced by Erd˝ os and studied by others. EG acbc ccba is an abelian square. It contributes to f 3 (4) . L. B. Richmond and J. Shallit Counting abelian squares The Electronic Journal of Combinatorics, Vol. 16, 2009. Arithmetic aspects of short random walks Armin Straub 12 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k � 1 � With k = 1 1! Γ 2 (3 / 2) = 4 1 2 : = (1 / 2)! 2 = 1 / 2 π Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k � 1 � With k = 1 1! Γ 2 (3 / 2) = 4 1 2 : = (1 / 2)! 2 = 1 / 2 π THM If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and Carlson f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k � 1 � With k = 1 1! Γ 2 (3 / 2) = 4 1 2 : = (1 / 2)! 2 = 1 / 2 π THM If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and Carlson f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. | f ( z ) | � Ae α | z | , and | f ( iy ) | � Be β | y | for β < π Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k � 1 � With k = 1 1! Γ 2 (3 / 2) = 4 1 2 : = (1 / 2)! 2 = 1 / 2 π THM If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and Carlson f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. • W n ( s ) is nice! | f ( z ) | � Ae α | z | , and | f ( iy ) | � Be β | y | for β < π Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a two-step walk EG W 2 (2 k ) : abelian squares of length 2 k from 2 letters b a b a a a b a a b � 2 k � Hence W 2 (2 k ) = . k � 1 � With k = 1 1! Γ 2 (3 / 2) = 4 1 2 : = (1 / 2)! 2 = 1 / 2 π THM If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and Carlson f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. • W n ( s ) is nice! � s � • Indeed, W 2 ( s ) = . | f ( z ) | � Ae α | z | , and s/ 2 | f ( iy ) | � Be β | y | for β < π Arithmetic aspects of short random walks Armin Straub 13 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � 1 � � 2 , − 1 2 , − 1 � 2 3 F 2 � 4 ≈ 1 . 574597238 − 0 . 126026522 i � 1 , 1 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) � 1 � � 2 , − 1 2 , − 1 � 2 3 F 2 � 4 ≈ 1 . 574597238 − 0 . 126026522 i � 1 , 1 Re ( W 3 ( s ) − V 3 ( s )) 0.05 2 4 6 8 10 � 0.05 � 0.10 � 0.15 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) � 1 � � 2 , − 1 2 , − 1 � 2 3 F 2 � 4 ≈ 1 . 574597238 − 0 . 126026522 i � 1 , 1 Re ( W 3 ( s ) − V 3 ( s )) 0.05 2 4 6 8 10 � 0.05 � 0.10 � 0.15 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) � 1 � � 2 , − 1 2 , − 1 � 2 3 F 2 � 4 ≈ 1 . 574597238 − 0 . 126026522 i � 1 , 1 Re ( W 3 ( s ) − V 3 ( s )) 0.05 | V 3 ( − i ( s + 1)) / V 3 ( − is ) | : e π = 23 . 1407 . . . 23.0 2 4 6 8 10 22.8 � 0.05 22.6 � 0.10 22.4 � 0.15 20 40 60 80 100 120 140 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) For integers k , THM � 1 � Borwein- � 2 , − k 2 , − k � Nuyens- 2 W 3 ( k ) = Re 3 F 2 � 4 . � S-Wan, 1 , 1 2010 Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a three-step walk � 1 k � � k � 2 � 2 j � � EG 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) For integers k , THM � 1 � Borwein- � 2 , − k 2 , − k � Nuyens- 2 W 3 ( k ) = Re 3 F 2 � 4 . � S-Wan, 1 , 1 2010 � 1 � � 2 � COR 2 1 / 3 2 2 / 3 W 3 (1) = 3 + 27 π 4 Γ 6 π 4 Γ 6 16 3 4 3 = 1 . 57459723755189 . . . Arithmetic aspects of short random walks Armin Straub 14 / 35

Moments of a four-step walk • Using Meijer G -function representations and transformations: � 5 � � THM 4 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 � W 4 ( − 1) = π Borwein- 2 � 4 7 F 6 � 1 S-Wan, � 1 4 , 1 , 1 , 1 , 1 , 1 2010 � 1 � 3 � � � � 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 3 2 , 3 2 , 3 2 , 3 2 , 3 = π + π � � 2 2 4 6 F 5 � 1 64 6 F 5 � 1 � � 1 , 1 , 1 , 1 , 1 2 , 2 , 2 , 2 , 2 � 2 n � 6 ∞ = π (4 n + 1) � n . 4 6 n 4 n =0 � 7 � � THM 4 , 3 2 , 3 2 , 3 2 , 1 2 , 1 2 , 1 W 4 (1) = 3 π � Borwein- 2 � 4 7 F 6 � 1 S-Wan, � 3 4 , 2 , 2 , 2 , 1 , 1 2010 � 7 � � 4 , 3 2 , 3 2 , 1 2 , 1 2 , 1 2 , 1 − 3 π � 2 � 8 7 F 6 � 1 . � 3 4 , 2 , 2 , 2 , 2 , 1 • We have no idea about the case of five steps. Arithmetic aspects of short random walks Armin Straub 15 / 35

A combinatorial convolution • From the interpretation as abelian squares: k � k � 2 � W n + m (2 k ) = W n (2 j ) W m (2( k − j )) . j j =0 Arithmetic aspects of short random walks Armin Straub 16 / 35

A combinatorial convolution • From the interpretation as abelian squares: k � k � 2 � W n + m (2 k ) = W n (2 j ) W m (2( k − j )) . j j =0 For even n , CONJ ∞ � s/ 2 � 2 � W n ( s ) ? = W n − 1 ( s − 2 j ) . j j =0 Arithmetic aspects of short random walks Armin Straub 16 / 35

A combinatorial convolution • From the interpretation as abelian squares: k � k � 2 � W n + m (2 k ) = W n (2 j ) W m (2( k − j )) . j j =0 For even n , CONJ ∞ � s/ 2 � 2 � W n ( s ) ? = W n − 1 ( s − 2 j ) . j j =0 • True for even s • True for n = 2 • True for n = 4 and integer s • In general, proven up to some technical growth conditions Arithmetic aspects of short random walks Armin Straub 16 / 35

Complex moments THM � � 2 k � W n (2 k ) = a 1 , . . . , a n a 1 + ··· + a n = k • Inevitable recursions K · f ( k ) = f ( k + 1) ( k + 2) 2 K 2 − (10 k 2 + 30 k + 23) K + 9( k + 1) 2 � � · W 3 (2 k ) = 0 ( k + 2) 3 K 2 − (2 k + 3)(10 k 2 + 30 k + 24) K + 64( k + 1) 3 � � · W 4 (2 k ) = 0 Arithmetic aspects of short random walks Armin Straub 17 / 35

Complex moments THM � � 2 k � W n (2 k ) = a 1 , . . . , a n a 1 + ··· + a n = k • Inevitable recursions K · f ( k ) = f ( k + 1) ( k + 2) 2 K 2 − (10 k 2 + 30 k + 23) K + 9( k + 1) 2 � � · W 3 (2 k ) = 0 ( k + 2) 3 K 2 − (2 k + 3)(10 k 2 + 30 k + 24) K + 64( k + 1) 3 � � · W 4 (2 k ) = 0 • Via Carlson’s Theorem these become functional equations 2 • W 3 ( s ) has a simple pole at − 2 with residue 3 π ; others at − 2 k . √ 4 4 3 3 2 2 1 1 � 6 � 4 � 2 2 � 6 � 4 � 2 2 � 1 � 1 � 2 � 2 � 3 � 3 Arithmetic aspects of short random walks Armin Straub 17 / 35

W 4 ( s ) in the complex plane 4 3 2 1 � 6 � 4 � 2 2 � 1 � 2 � 3 Arithmetic aspects of short random walks Armin Straub 18 / 35

W 4 ( s ) in the complex plane 4 3 2 1 � 6 � 4 � 2 2 � 1 � 2 � 3 Arithmetic aspects of short random walks Armin Straub 18 / 35

Crashcourse on the Mellin transform • Mellin transform F ( s ) of f ( x ) : W n ( s − 1) = M [ p n ; s ] � ∞ x s f ( x )d x M [ f ; s ] = x 0 Arithmetic aspects of short random walks Armin Straub 19 / 35

Crashcourse on the Mellin transform • Mellin transform F ( s ) of f ( x ) : W n ( s − 1) = M [ p n ; s ] � ∞ x s f ( x )d x M [ f ; s ] = x 0 • F ( s ) is analytic in a strip Thus functional equations • Functional properties: for F ( s ) translate into DEs • M [ x µ f ( x ); s ] = F ( s + µ ) for f ( x ) • M [ D x f ( x ); s ] = − ( s − 1) F ( s − 1) • M [ − θ x f ( x ); s ] = sF ( s ) Arithmetic aspects of short random walks Armin Straub 19 / 35

Crashcourse on the Mellin transform • Mellin transform F ( s ) of f ( x ) : W n ( s − 1) = M [ p n ; s ] � ∞ x s f ( x )d x M [ f ; s ] = x 0 • F ( s ) is analytic in a strip Thus functional equations • Functional properties: for F ( s ) translate into DEs • M [ x µ f ( x ); s ] = F ( s + µ ) for f ( x ) • M [ D x f ( x ); s ] = − ( s − 1) F ( s − 1) • M [ − θ x f ( x ); s ] = sF ( s ) • Poles of F ( s ) left of strip = ⇒ asymptotics of f ( x ) at zero ( − 1) n 1 n ! x m (log x ) n ( s + m ) n +1 Arithmetic aspects of short random walks Armin Straub 19 / 35

Mellin approach illustrated for p 2 0.8 0.6 0.4 � 2 k � • W 2 (2 k ) = 0.2 k 0.5 1.0 1.5 2.0 ( s + 2) W 2 ( s + 2) − 4( s + 1) W 2 ( s ) = 0 � x 2 ( θ x + 1) − 4 θ x � · p 2 ( x ) = 0 Arithmetic aspects of short random walks Armin Straub 20 / 35

Mellin approach illustrated for p 2 0.8 0.6 0.4 � 2 k � • W 2 (2 k ) = 0.2 k 0.5 1.0 1.5 2.0 ( s + 2) W 2 ( s + 2) − 4( s + 1) W 2 ( s ) = 0 � x 2 ( θ x + 1) − 4 θ x � · p 2 ( x ) = 0 C • Hence: p 2 ( x ) = √ 4 − x 2 Arithmetic aspects of short random walks Armin Straub 20 / 35

Mellin approach illustrated for p 2 0.8 0.6 0.4 � 2 k � • W 2 (2 k ) = 0.2 k 0.5 1.0 1.5 2.0 ( s + 2) W 2 ( s + 2) − 4( s + 1) W 2 ( s ) = 0 � x 2 ( θ x + 1) − 4 θ x � · p 2 ( x ) = 0 C • Hence: p 2 ( x ) = √ 4 − x 2 W 2 ( s ) = 1 1 s + 1 + O (1) as s → − 1 π p 2 ( x ) = 1 π + O ( x ) as x → 0 + 2 • Taken together: p 2 ( x ) = √ 4 − x 2 π Arithmetic aspects of short random walks Armin Straub 20 / 35

p 3 in hypergeometric form 0.7 0.6 0.5 0.4 • W 3 ( s ) has simple poles at − 2 k − 2 with residue 0.3 0.2 0.1 0.5 1.0 1.5 2.0 2.5 3.0 2 W 3 (2 k ) √ 3 2 k π 3 � x � ∞ � 2 k 2 x p 3 ( x ) = k =0 W 3 (2 k ) √ for 0 � x � 1 3 π 3 Arithmetic aspects of short random walks Armin Straub 21 / 35

p 3 in hypergeometric form 0.7 0.6 0.5 0.4 • W 3 ( s ) has simple poles at − 2 k − 2 with residue 0.3 0.2 0.1 0.5 1.0 1.5 2.0 2.5 3.0 2 W 3 (2 k ) √ 3 2 k π 3 � x � ∞ � 2 k 2 x p 3 ( x ) = k =0 W 3 (2 k ) √ for 0 � x � 1 3 π 3 � 2 � 2 j � k � • W 3 (2 k ) = � k is an Ap´ ery-like sequence j =0 j j Arithmetic aspects of short random walks Armin Straub 21 / 35

p 3 in hypergeometric form 0.7 0.6 0.5 0.4 • W 3 ( s ) has simple poles at − 2 k − 2 with residue 0.3 0.2 0.1 0.5 1.0 1.5 2.0 2.5 3.0 2 W 3 (2 k ) √ 3 2 k π 3 � x � ∞ � 2 k 2 x p 3 ( x ) = k =0 W 3 (2 k ) √ for 0 � x � 1 3 π 3 � 2 � 2 j � k � • W 3 (2 k ) = � k is an Ap´ ery-like sequence j =0 j j √ � � 3; 1; x 2 � 9 − x 2 � 2 2 3 x 3 , 2 1 p 3 ( x ) = π (3 + x 2 ) 2 F 1 (3 + x 2 ) 3 • Easy to verify once found • Holds for 0 � x � 3 Arithmetic aspects of short random walks Armin Straub 21 / 35

p 4 and its differential equation 0.5 0.4 0.3 0.2 0.1 1 2 3 4 � ( s + 4) 3 S 4 − 4( s + 3)(5 s 2 + 30 s + 48) S 2 + 64( s + 2) 3 � · W 4 ( s ) = 0 translates into A 4 · p 4 ( x ) = 0 with A 4 = x 4 ( θ x + 1) 3 − 4 x 2 θ x (5 θ 2 x + 3) + 64( θ x − 1) 3 Arithmetic aspects of short random walks Armin Straub 22 / 35

p 4 and its differential equation 0.5 0.4 0.3 0.2 0.1 1 2 3 4 � ( s + 4) 3 S 4 − 4( s + 3)(5 s 2 + 30 s + 48) S 2 + 64( s + 2) 3 � · W 4 ( s ) = 0 translates into A 4 · p 4 ( x ) = 0 with A 4 = x 4 ( θ x + 1) 3 − 4 x 2 θ x (5 θ 2 x + 3) + 64( θ x − 1) 3 p 4 ( x ) ≈ C √ 4 − x as x → 4 − . Thus p ′′ !! 4 is not locally integrable Care and does not have a Mellin transform in the classical sense. needed Arithmetic aspects of short random walks Armin Straub 22 / 35

p 4 and its differential equation 0.5 0.4 0.3 0.2 0.1 1 2 3 4 � ( s + 4) 3 S 4 − 4( s + 3)(5 s 2 + 30 s + 48) S 2 + 64( s + 2) 3 � · W 4 ( s ) = 0 translates into A 4 · p 4 ( x ) = 0 with A 4 = x 4 ( θ x + 1) 3 − 4 x 2 θ x (5 θ 2 x + 3) + 64( θ x − 1) 3 x + 6 x 4 � x 2 − 10 � = ( x − 4)( x − 2) x 3 ( x + 2)( x + 4) D 3 D 2 x 7 x 4 − 32 x 2 + 64 x 2 − 8 x 2 + 8 � � � � � � + x D x + p 4 ( x ) ≈ C √ 4 − x as x → 4 − . Thus p ′′ !! 4 is not locally integrable Care and does not have a Mellin transform in the classical sense. needed Arithmetic aspects of short random walks Armin Straub 22 / 35

Densities in general • The density p n satisfies a DE of order n − 1 . THM Borwein- • p n is real analytic except at 0 and the integers S-Wan- Zudilin, n, n − 2 , n − 4 , . . . . 2011 Arithmetic aspects of short random walks Armin Straub 23 / 35

Densities in general • The density p n satisfies a DE of order n − 1 . THM Borwein- • p n is real analytic except at 0 and the integers S-Wan- Zudilin, n, n − 2 , n − 4 , . . . . 2011 The second statement relies on an explicit recursion by Verrill (2004) as well as the combinatorial identity j j � � � � ( n − 2 m i ) 2 = α i ( n + 1 − α i ) . i =1 i =1 0 � m 1 ,...,mj<n/ 2 1 � α 1 ,...,αj � n mi<mi +1 αi � αi +1 − 2 First proven by Djakov-Mityagin (2004). Direct combinatorial proof by Zagier. Arithmetic aspects of short random walks Armin Straub 23 / 35

Densities in general • The density p n satisfies a DE of order n − 1 . THM EG Borwein- • p n is real analytic except at 0 and the integers n/ 2 − 1 S-Wan- n � n + 2 � � � ( n − 2 m ) 2 = Zudilin, n, n − 2 , n − 4 , . . . . α ( n + 1 − α ) = 2011 3 m =0 α =1 n/ 2 − 1 m 1 − 1 n α 1 − 2 The second statement relies on an explicit recursion by Verrill (2004) as � � � � ( n − 2 m 1 ) 2 ( n − 2 m 2 ) 2 = α 1 ( n + 1 − α 1 ) α 2 ( n + 1 − α 2 ) well as the combinatorial identity m 1 =0 m 2 =0 α 1 =1 α 2 =1 j j � � � � ( n − 2 m i ) 2 = α i ( n + 1 − α i ) . i =1 i =1 0 � m 1 ,...,mj<n/ 2 1 � α 1 ,...,αj � n mi<mi +1 αi � αi +1 − 2 First proven by Djakov-Mityagin (2004). Direct combinatorial proof by Zagier. Arithmetic aspects of short random walks Armin Straub 23 / 35

p 4 and its asymptotics at zero EG 3 ( s + 2) 2 + 9 log 2 1 1 W 4 ( s ) = s + 2 + O (1) as s → − 2 2 π 2 2 π 2 p 4 ( x ) = − 3 2 π 2 x log( x ) + 9 log 2 2 π 2 x + O ( x 3 ) as x → 0 + • W 4 ( s ) has double poles: s 4 ,k r 4 ,k W 4 ( s ) = ( s + 2 k + 2) 2 + s + 2 k + 2 + O (1) as s → − 2 k − 2 Arithmetic aspects of short random walks Armin Straub 24 / 35

p 4 and its asymptotics at zero EG 3 ( s + 2) 2 + 9 log 2 1 1 W 4 ( s ) = s + 2 + O (1) as s → − 2 2 π 2 2 π 2 p 4 ( x ) = − 3 2 π 2 x log( x ) + 9 log 2 2 π 2 x + O ( x 3 ) as x → 0 + • W 4 ( s ) has double poles: s 4 ,k r 4 ,k W 4 ( s ) = ( s + 2 k + 2) 2 + s + 2 k + 2 + O (1) as s → − 2 k − 2 ∞ � ( r 4 ,k − s 4 ,k log( x )) x 2 k +1 p 4 ( x ) = for small x � 0 k =0 Arithmetic aspects of short random walks Armin Straub 24 / 35

p 4 and its asymptotics at zero EG 3 ( s + 2) 2 + 9 log 2 1 1 W 4 ( s ) = s + 2 + O (1) as s → − 2 2 π 2 2 π 2 p 4 ( x ) = − 3 2 π 2 x log( x ) + 9 log 2 2 π 2 x + O ( x 3 ) as x → 0 + • W 4 ( s ) has double poles: s 4 ,k r 4 ,k W 4 ( s ) = ( s + 2 k + 2) 2 + s + 2 k + 2 + O (1) as s → − 2 k − 2 ∞ � ( r 4 ,k − s 4 ,k log( x )) x 2 k +1 p 4 ( x ) = for small x � 0 k =0 �� k � k � 2 � 2 j �� 2 n − 2 j � 3 W 4 (2 k ) � s 4 ,k = W 4 (2 k ) = 2 π 2 8 2 k j j n − j j =0 Domb numbers r 4 ,k known recursively Arithmetic aspects of short random walks Armin Straub 24 / 35

The Domb numbers • y 0 ( z ) := � k � 0 W 4 (2 k ) z k is the analytic solution of � 64 z 2 ( θ + 1) 3 − 2 z (2 θ + 1)(5 θ 2 + 5 θ + 2) + θ 3 � · y ( z ) = 0 . (DE) • Let y 1 ( z ) solve (DE) and y 1 ( z ) − y 0 ( z ) log( z ) ∈ z Q [[ z ]] . Then p 4 ( x ) = − 3 x 4 π 2 y 1 ( x 2 / 64) . Arithmetic aspects of short random walks Armin Straub 25 / 35

The Domb numbers • y 0 ( z ) := � k � 0 W 4 (2 k ) z k is the analytic solution of � 64 z 2 ( θ + 1) 3 − 2 z (2 θ + 1)(5 θ 2 + 5 θ + 2) + θ 3 � · y ( z ) = 0 . (DE) • Let y 1 ( z ) solve (DE) and y 1 ( z ) − y 0 ( z ) log( z ) ∈ z Q [[ z ]] . Then p 4 ( x ) = − 3 x 4 π 2 y 1 ( x 2 / 64) . Generating function for Domb numbers: THM � 1 Chan- ∞ � � 3 , 1 2 , 2 108 z 2 1 � � Chan-Liu W 4 (2 k ) z k = 3 1 − 4 z 3 F 2 � 2004; (1 − 4 z ) 3 1 , 1 � Rogers k =0 2009 Arithmetic aspects of short random walks Armin Straub 25 / 35

The Domb numbers • y 0 ( z ) := � k � 0 W 4 (2 k ) z k is the analytic solution of � 64 z 2 ( θ + 1) 3 − 2 z (2 θ + 1)(5 θ 2 + 5 θ + 2) + θ 3 � · y ( z ) = 0 . (DE) • Let y 1 ( z ) solve (DE) and y 1 ( z ) − y 0 ( z ) log( z ) ∈ z Q [[ z ]] . Then p 4 ( x ) = − 3 x 4 π 2 y 1 ( x 2 / 64) . Generating function for Domb numbers: THM � 1 Chan- ∞ � � 3 , 1 2 , 2 108 z 2 1 � � Chan-Liu W 4 (2 k ) z k = 3 1 − 4 z 3 F 2 � 2004; (1 − 4 z ) 3 1 , 1 � Rogers k =0 2009 � 1 � � 3 , 1 2 , 2 � • Basis at ∞ for the hypergeometric equation of 3 F 2 � t : 3 1 , 1 [as x → 4 then z = x 2 108 z 2 64 → 1 4 and t = (1 − 4 z ) 3 → ∞ ] � � � � � � � � � 1 3 , 1 3 , 1 2 , 1 1 2 , 1 3 , 2 2 3 , 2 � � � 1 1 1 t − 1 / 33 F 2 t − 1 / 23 F 2 t − 2 / 33 F 2 , , 3 2 3 � � � 3 , 5 2 t 5 6 , 7 t 4 3 , 7 t � � � 6 6 6 Arithmetic aspects of short random walks Armin Straub 25 / 35

p 4 in hypergeometric form 0.5 0.4 0.3 0.2 0.1 For 2 � x � 4 , THM 1 2 3 4 Borwein- √ S-Wan- � � � � 16 − x 2 � 3 1 2 , 1 2 , 1 p 4 ( x ) = 2 16 − x 2 � Zudilin 2 � 3 F 2 . 2011 � π 2 5 6 , 7 108 x 4 x � 6 • Easily (if tediously) provable once found Arithmetic aspects of short random walks Armin Straub 26 / 35

p 4 in hypergeometric form 0.5 0.4 0.3 0.2 0.1 For 2 � x � 4 , THM 1 2 3 4 Borwein- √ S-Wan- � � � � 16 − x 2 � 3 2 , 1 1 2 , 1 p 4 ( x ) = 2 16 − x 2 � Zudilin 2 � 3 F 2 . 2011 � π 2 6 , 7 5 108 x 4 x � 6 • Easily (if tediously) provable once found • Quite marvelously, as first observed numerically: For 0 � x � 4 , THM Borwein- √ S-Wan- � � � � 16 − x 2 � 3 1 2 , 1 2 , 1 p 4 ( x ) = 2 16 − x 2 � Zudilin 2 � Re 3 F 2 . 2011 � π 2 6 , 7 5 108 x 4 x � 6 Arithmetic aspects of short random walks Armin Straub 26 / 35

The density of a five-step random walk, again p 5 ( x ) = 0 . 32993 x +0 . 0066167 x 3 +0 . 00026233 x 5 +0 . 000014119 x 7 + O ( x 9 ) � ∞ 0.35 xtJ 0 ( xt ) J 5 p 5 ( x ) = 0 ( t ) d t 0.30 0 0.25 0.20 0.15 0.10 0.05 1 2 3 4 5 “ . . . the graphical construction, however carefully reinvestigated, did not permit of our considering the curve to be anything but a straight line. . . Even if it is not absolutely true, it exemplifies the extraordinary power of such integrals of J products to give extremely close approximations to ” such simple forms as horizontal lines. Karl Pearson , 1906 Arithmetic aspects of short random walks Armin Straub 27 / 35

The density of a five-step random walk, again x +0 . 0066167 x 3 +0 . 00026233 x 5 +0 . 000014119 x 7 + O ( x 9 ) p 5 ( x ) = 0 . 32993 = p 4 (1) � ∞ 0.35 xtJ 0 ( xt ) J 5 p 5 ( x ) = 0 ( t ) d t 0.30 0 0.25 0.20 0.15 0.10 0.05 1 2 3 4 5 “ . . . the graphical construction, however carefully reinvestigated, did not permit of our considering the curve to be anything but a straight line. . . Even if it is not absolutely true, it exemplifies the extraordinary power of such integrals of J products to give extremely close approximations to ” such simple forms as horizontal lines. Karl Pearson , 1906 Arithmetic aspects of short random walks Armin Straub 27 / 35

Modular differential equations Let f ( τ ) be a modular form and x ( τ ) a modular function w.r.t. Γ . THM • Then y ( x ) defined by f ( τ ) = y ( x ( τ )) satisfies a linear DE. • If x ( τ ) is a Hauptmodul for Γ , then the DE has polynomial coefficients. • The solutions of the DE are y ( x ) , τy ( x ) , τ 2 y ( x ) , . . . . Arithmetic aspects of short random walks Armin Straub 28 / 35

Modular differential equations Let f ( τ ) be a modular form and x ( τ ) a modular function w.r.t. Γ . THM • Then y ( x ) defined by f ( τ ) = y ( x ( τ )) satisfies a linear DE. • If x ( τ ) is a Hauptmodul for Γ , then the DE has polynomial coefficients. • The solutions of the DE are y ( x ) , τy ( x ) , τ 2 y ( x ) , . . . . ∞ � • Dedekind eta function: η ( τ ) = q 1 / 24 (1 − q n ) , q = e 2 πiτ n =1 � 1 / 2 , 1 / 2 � � EG � = θ 3 ( τ ) 2 2 F 1 � λ ( τ ) Classic � 1 • λ ( τ ) = 16 η ( τ/ 2) 8 η (2 τ ) 16 is the elliptic lambda function, a η ( τ ) 24 Hauptmodul for Γ(2) . η ( τ ) 5 • θ 3 ( τ ) = η ( τ/ 2) 2 η (2 τ ) 2 is the usual Jacobi theta function. Arithmetic aspects of short random walks Armin Straub 28 / 35

Modular differential equations Let f ( τ ) be a modular form and x ( τ ) a modular function w.r.t. Γ . THM • Then y ( x ) defined by f ( τ ) = y ( x ( τ )) satisfies a linear DE. • If x ( τ ) is a Hauptmodul for Γ , then the DE has polynomial coefficients. • The solutions of the DE are y ( x ) , τy ( x ) , τ 2 y ( x ) , . . . . ∞ � • Dedekind eta function: η ( τ ) = q 1 / 24 (1 − q n ) , q = e 2 πiτ n =1 � η (2 τ ) η (6 τ ) � 6 EG f ( τ ) = ( η ( τ ) η (3 τ )) 4 Chan- x ( τ ) = − , Chan-Liu ( η (2 τ ) η (6 τ )) 2 η ( τ ) η (3 τ ) 2004 = − q − 6 q 2 − 21 q 3 − 68 q 4 + . . . = 1 − 4 q + 4 q 2 − 4 q 3 + 20 q 4 + . . . � �� � 3 − 2 1 Here, Γ = Γ 0 (6) , . √ 6 − 3 3 Arithmetic aspects of short random walks Armin Straub 28 / 35

Modular differential equations Let f ( τ ) be a modular form and x ( τ ) a modular function w.r.t. Γ . THM • Then y ( x ) defined by f ( τ ) = y ( x ( τ )) satisfies a linear DE. • If x ( τ ) is a Hauptmodul for Γ , then the DE has polynomial coefficients. • The solutions of the DE are y ( x ) , τy ( x ) , τ 2 y ( x ) , . . . . ∞ � • Dedekind eta function: η ( τ ) = q 1 / 24 (1 − q n ) , q = e 2 πiτ n =1 � η (2 τ ) η (6 τ ) � 6 EG f ( τ ) = ( η ( τ ) η (3 τ )) 4 Chan- x ( τ ) = − , Chan-Liu ( η (2 τ ) η (6 τ )) 2 η ( τ ) η (3 τ ) 2004 = − q − 6 q 2 − 21 q 3 − 68 q 4 + . . . = 1 − 4 q + 4 q 2 − 4 q 3 + 20 q 4 + . . . � �� � 3 − 2 1 Here, Γ = Γ 0 (6) , . Then, in a neighborhood of i ∞ , √ 6 − 3 3 ∞ � W 4 (2 k ) x ( τ ) k . f ( τ ) = y 0 ( x ( τ )) = k =0 Arithmetic aspects of short random walks Armin Straub 28 / 35

Modular parametrization of p 4 THM For τ = − 1 / 2 + iy and y > 0 : Borwein- S-Wan- � � η (2 τ ) η (6 τ ) � 3 � = 6(2 τ + 1) Zudilin p 4 8 i η ( τ ) η (2 τ ) η (3 τ ) η (6 τ ) 2011 η ( τ ) η (3 τ ) π = √ = √ − x ( τ ) f ( τ ) 64 x ( τ ) Arithmetic aspects of short random walks Armin Straub 29 / 35

Modular parametrization of p 4 THM For τ = − 1 / 2 + iy and y > 0 : Borwein- S-Wan- � � η (2 τ ) η (6 τ ) � 3 � = 6(2 τ + 1) Zudilin p 4 8 i η ( τ ) η (2 τ ) η (3 τ ) η (6 τ ) 2011 η ( τ ) η (3 τ ) π = √ = √ − x ( τ ) f ( τ ) 64 x ( τ ) √− 15 , one obtains p 4 (1) = p ′ • When τ = − 1 2 + 1 5 (0) as an η -product. 6 Arithmetic aspects of short random walks Armin Straub 29 / 35

Modular parametrization of p 4 THM For τ = − 1 / 2 + iy and y > 0 : Borwein- S-Wan- � � η (2 τ ) η (6 τ ) � 3 � = 6(2 τ + 1) Zudilin p 4 8 i η ( τ ) η (2 τ ) η (3 τ ) η (6 τ ) 2011 η ( τ ) η (3 τ ) π = √ = √ − x ( τ ) f ( τ ) 64 x ( τ ) √− 15 , one obtains p 4 (1) = p ′ • When τ = − 1 2 + 1 5 (0) as an η -product. 6 • Applying the Chowla–Selberg formula, eventually leads to: √ COR 5 p 4 (1) = p ′ 40 π 4 Γ( 1 15 )Γ( 2 15 )Γ( 4 15 )Γ( 8 5 (0) = 15 ) ≈ 0 . 32993 Arithmetic aspects of short random walks Armin Straub 29 / 35

Chowla–Selberg formula   3 w THM | d | h � ( d Chowla– 1 � k ) � � j | η ( τ j ) | 24 = a − 6 k Selberg Γ   | d | 1967 (2 π | d | ) 6 h j =1 k =1 where the product is over reduced binary quadratic forms √ [ a j , b j , c j ] of discriminant d < 0 . Further, τ j = − b j + d . 2 a j Arithmetic aspects of short random walks Armin Straub 30 / 35

Chowla–Selberg formula   3 w THM | d | h � ( d Chowla– 1 � k ) � � j | η ( τ j ) | 24 = a − 6 k Selberg Γ   | d | 1967 (2 π | d | ) 6 h j =1 k =1 where the product is over reduced binary quadratic forms √ [ a j , b j , c j ] of discriminant d < 0 . Further, τ j = − b j + d . 2 a j Q ( √− 15) has discriminant ∆ = − 15 and class number h = 2 . EG Q 1 = [1 , 1 , 4] , Q 2 = [2 , 1 , 2] with corresponding roots √ τ 1 = − 1 2 + 1 τ 2 = 1 − 15 , 2 τ 1 . 2 � � 1 / 2 Γ( 1 15 )Γ( 2 15 )Γ( 4 15 )Γ( 8 1 1 15 ) 2 | η ( τ 1 ) η ( τ 2 ) | 2 = √ Γ( 7 15 )Γ( 11 15 )Γ( 13 15 )Γ( 14 30 π 15 ) 1 120 π 3 Γ( 1 15 )Γ( 2 15 )Γ( 4 15 )Γ( 8 = 15 ) Arithmetic aspects of short random walks Armin Straub 30 / 35

Evaluating eta-quotients √ Fact If σ 1 , σ 2 ∈ H both belong to Q ( − d ) , then the quotient η ( σ 1 ) /η ( σ 2 ) is an algebraic number. Arithmetic aspects of short random walks Armin Straub 31 / 35

Evaluating eta-quotients √ Fact If σ 1 , σ 2 ∈ H both belong to Q ( − d ) , then the quotient η ( σ 1 ) /η ( σ 2 ) is an algebraic number. • We can write σ 2 = M · σ 1 for some M ∈ GL 2 ( Z ) . Proof. η ( τ ) • f ( τ ) = η ( M · τ ) is a modular function. Arithmetic aspects of short random walks Armin Straub 31 / 35

Evaluating eta-quotients √ Fact If σ 1 , σ 2 ∈ H both belong to Q ( − d ) , then the quotient η ( σ 1 ) /η ( σ 2 ) is an algebraic number. • We can write σ 2 = M · σ 1 for some M ∈ GL 2 ( Z ) . Proof. η ( τ ) • f ( τ ) = η ( M · τ ) is a modular function. • σ 1 = N · σ 1 for some non-identity N ∈ GL 2 ( Z ) . • f ( N · τ ) is another modular function. Arithmetic aspects of short random walks Armin Straub 31 / 35

Evaluating eta-quotients √ Fact If σ 1 , σ 2 ∈ H both belong to Q ( − d ) , then the quotient η ( σ 1 ) /η ( σ 2 ) is an algebraic number. • We can write σ 2 = M · σ 1 for some M ∈ GL 2 ( Z ) . Proof. η ( τ ) • f ( τ ) = η ( M · τ ) is a modular function. • σ 1 = N · σ 1 for some non-identity N ∈ GL 2 ( Z ) . • f ( N · τ ) is another modular function. • There is an algebraic relation Φ( f ( τ ) , f ( N · τ )) = 0 . Arithmetic aspects of short random walks Armin Straub 31 / 35

Evaluating eta-quotients √ Fact If σ 1 , σ 2 ∈ H both belong to Q ( − d ) , then the quotient η ( σ 1 ) /η ( σ 2 ) is an algebraic number. • We can write σ 2 = M · σ 1 for some M ∈ GL 2 ( Z ) . Proof. η ( τ ) • f ( τ ) = η ( M · τ ) is a modular function. • σ 1 = N · σ 1 for some non-identity N ∈ GL 2 ( Z ) . • f ( N · τ ) is another modular function. • There is an algebraic relation Φ( f ( τ ) , f ( N · τ )) = 0 . • Then: Φ( f ( σ 1 ) , f ( σ 1 )) = 0 Arithmetic aspects of short random walks Armin Straub 31 / 35