Special values of trigonometric Dirichlet series Legacy of - PowerPoint PPT Presentation

Special values of trigonometric Dirichlet series Legacy of Ramanujan OPSFA-13, NIST Armin Straub June 3, 2015 University of Illinois at Urbana–Champaign Includes joint work with : √ ∞ sec 2 ( πn 5) = 14 � 135 π 4 n 4 n =1 √ ∞ tan 3 ( πn 6) 35 , 159 � 6 π 4 = √ n 5 17 , 820 n =1 Bruce Berndt University of Illinois at Urbana–Champaign Special values of trigonometric Dirichlet series Armin Straub 1 / 17

Rough outline • examples of special values of trigonometric Dirichlet series • main result on special values and outline of strategy • just a brief comment on convergence • introduction to Eichler integrals of Eisenstein series • open problems (possibly unimodularity, if time permits) Special values of trigonometric Dirichlet series Armin Straub 2 / 17

Basic examples of trigonometric Dirichlet series • Euler’s identity: ∞ n 2 m = − 1 1 2(2 πi ) 2 m B 2 m � (2 m )! n =1 Special values of trigonometric Dirichlet series Armin Straub 3 / 17

Basic examples of trigonometric Dirichlet series • Euler’s identity: ∞ n 2 m = − 1 1 2(2 πi ) 2 m B 2 m � (2 m )! n =1 • Half of the Clausen and Glaisher functions reduce, e.g., ∞ poly 1 ( τ ) = π 2 cos( πnτ ) 3 τ 2 − 6 τ + 2 � � � = poly m ( τ ) , . n 2 m 12 n =1 Special values of trigonometric Dirichlet series Armin Straub 3 / 17

Basic examples of trigonometric Dirichlet series • Euler’s identity: ∞ n 2 m = − 1 1 2(2 πi ) 2 m B 2 m � (2 m )! n =1 • Half of the Clausen and Glaisher functions reduce, e.g., ∞ poly 1 ( τ ) = π 2 cos( πnτ ) 3 τ 2 − 6 τ + 2 � � � = poly m ( τ ) , . n 2 m 12 n =1 • Ramanujan investigated trigonometric Dirichlet series of similar type. From his first letter to Hardy: ∞ = 19 π 7 coth( πn ) � n 7 56700 n =1 In fact, this was already included in a general formula by Lerch. Special values of trigonometric Dirichlet series Armin Straub 3 / 17

One of Ramanujan’s most famous formulas For α, β > 0 such that αβ = π 2 and m ∈ Z , THM Ramanujan, Grosswald � ∞ � � ∞ � n − 2 m − 1 n − 2 m − 1 ζ (2 m + 1) ζ (2 m + 1) α − m � = ( − β ) − m � + + e 2 αn − 1 e 2 βn − 1 2 2 n =1 n =1 m +1 ( − 1) n B 2 n B 2 m − 2 n +2 − 2 2 m � (2 m − 2 n + 2)! α m − n +1 β n . (2 n )! n =0 Special values of trigonometric Dirichlet series Armin Straub 4 / 17

One of Ramanujan’s most famous formulas For α, β > 0 such that αβ = π 2 and m ∈ Z , THM Ramanujan, Grosswald � ∞ � � ∞ � n − 2 m − 1 n − 2 m − 1 ζ (2 m + 1) ζ (2 m + 1) α − m � = ( − β ) − m � + + e 2 αn − 1 e 2 βn − 1 2 2 n =1 n =1 m +1 ( − 1) n B 2 n B 2 m − 2 n +2 − 2 2 m � (2 m − 2 n + 2)! α m − n +1 β n . (2 n )! n =0 1 e x − 1 = 1 2 cot( x 2 ) − 1 • In terms of ∞ 2 cot( πnτ ) � ξ s ( τ ) = , n s n =1 Ramanujan’s formula takes the form k B 2 s B 2 k − 2 s � τ 2 m − 2 ξ 2 m − 1 ( − 1 τ ) − ξ 2 m − 1 ( τ ) = ( − 1) k (2 π ) 2 k − 1 (2 k − 2 s )! τ 2 s − 1 . (2 s )! s =0 Special values of trigonometric Dirichlet series Armin Straub 4 / 17

One of Ramanujan’s most famous formulas For α, β > 0 such that αβ = π 2 and m ∈ Z , THM Ramanujan, Grosswald � ∞ � � ∞ � n − 2 m − 1 n − 2 m − 1 ζ (2 m + 1) ζ (2 m + 1) α − m � = ( − β ) − m � + + e 2 αn − 1 e 2 βn − 1 2 2 n =1 n =1 m +1 ( − 1) n B 2 n B 2 m − 2 n +2 − 2 2 m � (2 m − 2 n + 2)! α m − n +1 β n . (2 n )! n =0 1 e x − 1 = 1 2 cot( x 2 ) − 1 • In terms of ∞ 2 cot( πnτ ) � ξ s ( τ ) = , n s n =1 Ramanujan’s formula takes the form k B 2 s B 2 k − 2 s � τ 2 m − 2 ξ 2 m − 1 ( − 1 τ ) − ξ 2 m − 1 ( τ ) = ( − 1) k (2 π ) 2 k − 1 (2 k − 2 s )! τ 2 s − 1 . (2 s )! s =0 ∞ = 19 π 7 coth( πn ) � • Set m = 4 and τ = i to obtain 56700 . n 7 n =1 Special values of trigonometric Dirichlet series Armin Straub 4 / 17

Special values of trigonometric Dirichlet series ∞ ∞ ( − 1) n +1 csch( πn ) EG = π 3 = π 3 tanh((2 n + 1) π/ 2) � � Ramanujan 32 , (2 n + 1) 3 n 3 360 n =0 n =1 Special values of trigonometric Dirichlet series Armin Straub 5 / 17

Special values of trigonometric Dirichlet series ∞ ∞ ( − 1) n +1 csch( πn ) EG = π 3 = π 3 tanh((2 n + 1) π/ 2) � � Ramanujan 32 , (2 n + 1) 3 n 3 360 n =0 n =1 √ EG � � √ πn 1+ 5 ∞ cot ∞ = − π 3 23 π 5 tan( π (2 n + 1) 5) Berndt 2 � � √ 5 , = √ 1976-78 n 3 (2 n + 1) 5 45 3456 5 n =1 n =0 Let τ = ( a + b √ c ) / 2 for a, b, c ∈ Q with c > 0 and a 2 − cb 2 = 4 ε , THM Berndt ε = ± 1 . If k > 1 , 1976 ∞ k = ( − 1) k − 1 (2 π ) 2 k − 1 cot( πnτ ) B 2 s B 2 k − 2 s � � (2 k − 2 s )! τ 2 s − 1 . n 2 k − 1 1 − ετ 2 k − 2 (2 s )! n =1 s =0 Special values of trigonometric Dirichlet series Armin Straub 5 / 17

Special values of trigonometric Dirichlet series ∞ ∞ ( − 1) n +1 csch( πn ) EG = π 3 = π 3 tanh((2 n + 1) π/ 2) � � Ramanujan 32 , (2 n + 1) 3 n 3 360 n =0 n =1 √ EG � � √ πn 1+ 5 ∞ cot ∞ = − π 3 23 π 5 tan( π (2 n + 1) 5) Berndt 2 � � √ 5 , = √ 1976-78 n 3 (2 n + 1) 5 45 3456 5 n =1 n =0 Let τ = ( a + b √ c ) / 2 for a, b, c ∈ Q with c > 0 and a 2 − cb 2 = 4 ε , THM Berndt ε = ± 1 . If k > 1 , 1976 ∞ k = ( − 1) k − 1 (2 π ) 2 k − 1 cot( πnτ ) B 2 s B 2 k − 2 s � � (2 k − 2 s )! τ 2 s − 1 . n 2 k − 1 1 − ετ 2 k − 2 (2 s )! n =1 s =0 EG ∞ ∞ cot 2 ( πnζ 3 ) csc 2 ( πnζ 3 ) = − 31 1 � 2835 π 4 , � 5670 π 4 Komori- = Matsumoto- n 4 n 4 Tsumura n =1 n =1 2013 (Here, ζ 3 is the primitive third root of unity.) Special values of trigonometric Dirichlet series Armin Straub 5 / 17

Secant zeta function • Lal´ ın, Rodrigue and Rogers introduce and study ∞ sec( πnτ ) � ψ s ( τ ) = . n s n =1 • Clearly, ψ s (0) = ζ ( s ) . In particular, ψ 2 (0) = π 2 6 . Special values of trigonometric Dirichlet series Armin Straub 6 / 17

Secant zeta function • Lal´ ın, Rodrigue and Rogers introduce and study ∞ sec( πnτ ) � ψ s ( τ ) = . n s n =1 • Clearly, ψ s (0) = ζ ( s ) . In particular, ψ 2 (0) = π 2 6 . EG √ √ 2) = − π 2 6) = 2 π 2 LRR ’13 ψ 2 ( 3 , ψ 2 ( 3 Special values of trigonometric Dirichlet series Armin Straub 6 / 17

Secant zeta function • Lal´ ın, Rodrigue and Rogers introduce and study ∞ sec( πnτ ) � ψ s ( τ ) = . n s n =1 • Clearly, ψ s (0) = ζ ( s ) . In particular, ψ 2 (0) = π 2 6 . EG √ √ 2) = − π 2 6) = 2 π 2 LRR ’13 ψ 2 ( 3 , ψ 2 ( 3 For positive integers m , r , CONJ LRR ’13 ψ 2 m ( √ r ) ∈ Q · π 2 m . • proof completed independently by Berndt–S and Charollois–Greenberg Special values of trigonometric Dirichlet series Armin Straub 6 / 17

Special values of trigonometric Dirichlet series √ ∞ EG sec 2 ( πn 5) = 14 � 135 π 4 S 2014 n 4 n =1 Special values of trigonometric Dirichlet series Armin Straub 7 / 17

Special values of trigonometric Dirichlet series √ ∞ EG sec 2 ( πn 5) = 14 � 135 π 4 S 2014 n 4 n =1 √ ∞ cot 2 ( πn 5) = 13 � 945 π 4 n 4 n =1 Special values of trigonometric Dirichlet series Armin Straub 7 / 17

Special values of trigonometric Dirichlet series √ ∞ EG sec 2 ( πn 5) = 14 � 135 π 4 S 2014 n 4 n =1 √ ∞ cot 2 ( πn 5) = 13 � 945 π 4 n 4 n =1 √ ∞ csc 2 ( πn 11) 8 � 385 π 4 = n 4 n =1 √ ∞ sec 3 ( πn 2) = − 2483 � 5220 π 4 n 4 n =1 √ ∞ tan 3 ( πn 6) 35 , 159 � 6 π 4 = √ n 5 17 , 820 n =1 Special values of trigonometric Dirichlet series Armin Straub 7 / 17

Special values of trigonometric Dirichlet series • For a, b ∈ Z , let trig a,b = sec a csc b be any product/quotient of trigonometric functions. ∞ THM trig a,b ( πnρ ) � ∈ π s Q ( ρ ) S 2014 n s n =1 provided that • ρ is a real quadratic irrationality, • s � max( a, b, 1) + 1 (so that the series converges), • s and b have the same parity. Special values of trigonometric Dirichlet series Armin Straub 8 / 17

Special values of trigonometric Dirichlet series • For a, b ∈ Z , let trig a,b = sec a csc b be any product/quotient of trigonometric functions. ∞ THM trig a,b ( πnρ ) � ∈ π s Q ( ρ ) S 2014 n s n =1 provided that • ρ is a real quadratic irrationality, • s � max( a, b, 1) + 1 (so that the series converges), • s and b have the same parity. • If, in addition, ρ 2 ∈ Q and a + b � 0 , then the value is in ( πρ ) s Q . √ ∞ EG (cos cot)( πn 2) � 1 253 � � π 3 √ = 2 − n 3 360 2 n =1 (Here, ( a, b ) = ( − 2 , 1) does not satisfy a + b � 0 .) Special values of trigonometric Dirichlet series Armin Straub 8 / 17

Strategy ∞ trig a,b ( πnρ ) • Rough strategy how to evaluate ψ a,b � s ( ρ ) = : n s ( trig a,b = sec a csc b ) n =1 Special values of trigonometric Dirichlet series Armin Straub 9 / 17