On a secant Dirichlet series and Eichler integrals of Eisenstein - PowerPoint PPT Presentation

On a secant Dirichlet series and Eichler integrals of Eisenstein series Modular Forms and Modular Integrals in Memory of Marvin Knopp AMS Sectional Meeting, Temple University Armin Straub October 12, 2013 University of Illinois & Max-Planck-Institut at Urbana–Champaign f¨ ur Mathematik, Bonn 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 / 22

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 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 2 / 22

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 / 22

Secant zeta function: Motivation • 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 ) = τ 2 2 + π 2 cos( nτ ) 4 − πτ � = poly m ( τ ) , 6 . n 2 m 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 / 22

Secant zeta function: Convergence • ψ s ( τ ) = � sec( πnτ ) has singularity at rationals with even denominator n s 10 6 4 5 2 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 � 5 � 2 � 4 � 10 ψ 2 ( τ ) truncated to 4 and 8 terms Re ψ 2 ( τ + εi ) with ε = 1 / 1000 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 4 / 22

Secant zeta function: Convergence • ψ s ( τ ) = � sec( πnτ ) has singularity at rationals with even denominator n s 10 6 4 5 2 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 � 5 � 2 � 4 � 10 ψ 2 ( τ ) truncated to 4 and 8 terms 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 4 / 22

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 5 / 22

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 5 / 22

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 5 / 22

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 ) � 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 5 / 22

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 ) � a b DEF � aτ + b � ( τ ) = ( cτ + d ) − k F � F | k slash c d cτ + d operator • In terms of � 1 � � 0 � � 1 � 1 − 1 0 T = , S = , R = , 0 1 1 0 1 1 the functional equations become ψ 2 m | 1 − 2 m ( T 2 − 1) = 0 , ψ 2 m | 1 − 2 m ( R 2 − 1) = π 2 m rat( τ ) . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 5 / 22

Secant zeta function: Functional equation • The matrices � 1 � � 1 � 2 0 T 2 = R 2 = , , 0 1 2 1 together with − I , generate Γ(2) = { γ ∈ SL 2 ( Z ) : γ ≡ I (mod 2) } . COR For any γ ∈ Γ(2) , ψ 2 m | 1 − 2 m ( γ − 1) = π 2 m rat( τ ) . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 6 / 22

Secant zeta function: Special values For positive integers m , r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 • Note that proof � X � · √ r = √ r. rY Y X • As shown by Lagrange, there are X and Y which solve Pell’s equation X 2 − rY 2 = 1 . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 7 / 22

Secant zeta function: Special values For positive integers m , r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 • Note that proof � X � · √ r = √ r. rY Y X • As shown by Lagrange, there are X and Y which solve Pell’s equation X 2 − rY 2 = 1 . • Since � 2 � X 2 + rY 2 � X � rY 2 rXY γ = = ∈ Γ(2) , X 2 + rY 2 Y X 2 XY the claim follows from the evenness of ψ 2 m and ψ 2 m | 1 − 2 m ( γ − 1) = π 2 m rat( τ ) . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 7 / 22

Eichler integrals • F is an Eichler integral if D k − 1 F is modular of weight k . • Such 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 / 22

Eichler integrals • F is an Eichler integral if D k − 1 F is modular of weight k . • Such 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 � i ∞ ˜ [ f ( z ) − a 0 ] ( z − τ ) k − 2 d z f ( τ ) = τ = ( − 1) k Γ( k − 1) ∞ a ( n ) � n k − 1 q n . (2 πi ) k − 1 n =1 If a 0 = 0 , ˜ f is an Eichler integral as defined above. On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 8 / 22

Eichler integrals of Eisenstein series • For the Eisenstein series G 2 k , G 2 k ( τ ) = 2 ζ (2 k ) + 2(2 πi ) 2 k ∞ � σ 2 k − 1 ( n ) q n , Γ(2 k ) n =1 � n 2 k − 1 q n 1 − q n ∞ 4 πi σ 2 k − 1 ( n ) q n . ˜ � G 2 k ( τ ) = n 2 k − 1 2 k − 1 n =1 � n 1 − 2 k q n 1 − q n On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 9 / 22

Eichler integrals of Eisenstein series • For the Eisenstein series G 2 k , G 2 k ( τ ) = 2 ζ (2 k ) + 2(2 πi ) 2 k ∞ � σ 2 k − 1 ( n ) q n , Γ(2 k ) n =1 � n 2 k − 1 q n 1 − q n ∞ 4 πi σ 2 k − 1 ( n ) q n . ˜ � G 2 k ( τ ) = n 2 k − 1 2 k − 1 n =1 � n 1 − 2 k q n 1 − q n • The period “polynomial” ˜ G 2 k | 2 − 2 k ( S − 1) is given by � k � (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 / 22