On a secant Dirichlet series and Eichler integrals of Eisenstein series 28th Automorphic Forms Workshop Moab, Utah Armin Straub May 12, 2014 University of Illinois at Urbana–Champaign Based on joint work with : Bruce Berndt University of Illinois at Urbana–Champaign On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 1 / 18
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 . √ 2) = − π 2 √ 6) = 2 π 2 EG LRR ’13 ψ 2 ( 3 , ψ 2 ( 3 For positive integers m , r , CONJ LRR ’13 ψ 2 m ( √ r ) ∈ Q · π 2 m . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 2 / 18
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. On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 3 / 18
One of Ramanujan’s most well-known 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 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 4 / 18
One of Ramanujan’s most well-known 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 • In terms of ξ s ( τ ) = � cot( πnτ ) , Ramanujan’s formula becomes n s k B 2 s B 2 k − 2 s ξ 2 k − 1 | 2 − 2 k ( S − 1) = ( − 1) k (2 π ) 2 k − 1 � (2 k − 2 s )! τ 2 s − 1 . (2 s )! s =0 � a b DEF � aτ + b � ( τ ) = ( cτ + d ) − k F � F | k slash c d cτ + d operator On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 4 / 18
Secant zeta function: Convergence • ψ s ( τ ) = � sec( πnτ ) has singularity at rationals with even denominator n s 10 5 5 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 � 5 � 5 � 10 Re ψ 2 ( τ + εi ) with ε = 1 / 100 Re ψ 2 ( τ + εi ) with ε = 1 / 1000 The series ψ s ( τ ) = � sec( πnτ ) THM converges absolutely if n s Lal´ ın– Rodrigue– 1 τ = p/q with q odd and s > 1 , Rogers 2013 2 τ is algebraic irrational and s � 2 . • Proof uses Thue–Siegel–Roth, as well as a result of Worley when s = 2 and τ is irrational On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 5 / 18
Secant zeta function: Functional equation • Obviously, ψ s ( τ ) = � sec( πnτ ) satisfies ψ s ( τ + 2) = ψ s ( τ ) . n s THM � τ � � τ � (1 + τ ) 2 m − 1 ψ 2 m − (1 − τ ) 2 m − 1 ψ 2 m LRR, BS 1 + τ 1 − τ 2013 = π 2 m rat( τ ) On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 6 / 18
Secant zeta function: Functional equation • Obviously, ψ s ( τ ) = � sec( πnτ ) satisfies ψ s ( τ + 2) = ψ s ( τ ) . n s THM � τ � � τ � (1 + τ ) 2 m − 1 ψ 2 m − (1 − τ ) 2 m − 1 ψ 2 m LRR, BS 1 + τ 1 − τ 2013 = π 2 m rat( τ ) Collect residues of the integral proof 1 � sin ( πτz ) d z I C = z s +1 . 2 πi sin( π (1 + τ ) z ) sin( π (1 − τ ) z ) C C are appropriate circles around the origin such that I C → 0 as radius( C ) → ∞ . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 6 / 18
Secant zeta function: Functional equation • Obviously, ψ s ( τ ) = � sec( πnτ ) satisfies ψ s ( τ + 2) = ψ s ( τ ) . n s THM � τ � � τ � (1 + τ ) 2 m − 1 ψ 2 m − (1 − τ ) 2 m − 1 ψ 2 m LRR, BS 1 + τ 1 − τ 2013 sin( τz ) = π 2 m [ z 2 m − 1 ] sin((1 − τ ) z ) sin((1 + τ ) z ) Collect residues of the integral proof 1 � sin ( πτz ) d z I C = z s +1 . 2 πi sin( π (1 + τ ) z ) sin( π (1 − τ ) z ) C C are appropriate circles around the origin such that I C → 0 as radius( C ) → ∞ . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 6 / 18
Secant zeta function: Functional equation • Obviously, ψ s ( τ ) = � sec( πnτ ) satisfies ψ s ( τ + 2) = ψ s ( τ ) . n s THM � τ � � τ � (1 + τ ) 2 m − 1 ψ 2 m − (1 − τ ) 2 m − 1 ψ 2 m LRR, BS 1 + τ 1 − τ 2013 sin( τz ) = π 2 m [ z 2 m − 1 ] sin((1 − τ ) z ) sin((1 + τ ) z ) � 1 � � 1 � 1 0 • In terms of T = and R = , 0 1 1 1 ψ 2 m | 1 − 2 m ( T 2 − 1) = 0 , ψ 2 m | 1 − 2 m ( R 2 − 1) = π 2 m rat( τ ) . ψ 2 m | 1 − 2 m ( γ − 1) = π 2 m rat( τ ) . COR For any γ ∈ Γ(2) , On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 6 / 18
Secant zeta function: Special values For positive integers m , r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 proof • Any real quadratic irrational τ is fixed by some γ ∈ Γ(2) . This follows from Pell’s equation. • Combined with ψ 2 m | 1 − 2 m ( γ − 1) = π 2 m rat( τ ) , it follows that ψ 2 m ( τ ) ∈ Q ( τ ) · π 2 m . • Finally, use the fact that ψ 2 m is even. On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 7 / 18
Eichler integrals • F is an Eichler integral if D k − 1 F is modular of weight k . D = q d d q EG ∞ ∞ ∞ ∞ n 2 k − 1 q n n 1 − 2 k q n σ 2 k − 1 ( n ) integrate σ 2 k − 1 ( n ) q n = q n = � � � � − − − − − → 1 − q n n 2 k − 1 1 − q n n =1 n =1 n =1 n =1 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 8 / 18
Eichler integrals • F is an Eichler integral if D k − 1 F is modular of weight k . D = q d d q EG ∞ ∞ ∞ ∞ n 2 k − 1 q n n 1 − 2 k q n σ 2 k − 1 ( n ) integrate σ 2 k − 1 ( n ) q n = q n = � � � � − − − − − → 1 − q n n 2 k − 1 1 − q n n =1 n =1 n =1 n =1 • Eichler integrals are characterized by F | 2 − k ( γ − 1) = poly( τ ) , deg poly � k − 2 . • poly( τ ) is a period polynomial of the modular form f . The period polynomial encodes the critical L -values of f . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 8 / 18
Eichler integrals • F is an Eichler integral if D k − 1 F is modular of weight k . D = q d d q EG ∞ ∞ ∞ ∞ n 2 k − 1 q n n 1 − 2 k q n σ 2 k − 1 ( n ) integrate σ 2 k − 1 ( n ) q n = q n = � � � � − − − − − → 1 − q n n 2 k − 1 1 − q n n =1 n =1 n =1 n =1 • Eichler integrals are characterized by F | 2 − k ( γ − 1) = poly( τ ) , deg poly � k − 2 . • poly( τ ) is a period polynomial of the modular form f . The period polynomial encodes the critical L -values of f . • For a modular form f ( τ ) = � a ( n ) q n of weight k , define ∞ f ( τ ) = ( − 1) k Γ( k − 1) a ( n ) ˜ � n k − 1 q n . (2 πi ) k − 1 n =1 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 8 / 18
Eichler integrals of Eisenstein series • For the Eisenstein series G 2 k , the period “polynomial” is � k � G 2 k | 2 − 2 k ( S − 1) = (2 πi ) 2 k B 2 s (2 k − 2 s )! X 2 s − 1 + ζ (2 k − 1) B 2 k − 2 s (2 πi ) 2 k − 1 ( X 2 k − 2 − 1) ˜ � . 2 k − 1 (2 s )! s =0 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 9 / 18
Eichler integrals of Eisenstein series • For the Eisenstein series G 2 k , the period “polynomial” is � k � G 2 k | 2 − 2 k ( S − 1) = (2 πi ) 2 k B 2 s (2 k − 2 s )! X 2 s − 1 + ζ (2 k − 1) B 2 k − 2 s (2 πi ) 2 k − 1 ( X 2 k − 2 − 1) ˜ � . 2 k − 1 (2 s )! s =0 • In other words, � cot( πnτ ) is an Eichler integral of G 2 k . n 2 k − 1 cot( πτ ) = 1 1 EG � π τ + j j ∈ Z N � lim N →∞ j = − N On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 9 / 18
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