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

On a secant Dirichlet series and Eichler integrals of Eisenstein series Oberseminar Zahlentheorie Universit¨ at zu K¨ oln Armin Straub November 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 / 33

PART I A secant Dirichlet series ∞ sec( πnτ ) � ψ s ( τ ) = n s n =1 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 2 / 33

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 3 / 33

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 3 / 33

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 4 / 33

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 5 / 33

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 5 / 33

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 / 33

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 / 33

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 / 33

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 6 / 33

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 6 / 33

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 7 / 33

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 . 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 8 / 33

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 8 / 33

Secant zeta function: Special values For integers κ, µ , EG � �� = π 2 � � 1 + 3 κ � � 1 ψ 2 κ + κ µ + κ , 6 2 µ � �� 2 µ − 5 κ 2 (16 µ 2 − 15) = π 4 � � 1 + 5 κ � � 1 ψ 4 κ + κ µ + κ . 8 µ 2 (4 κµ + 3) 90 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 9 / 33

PART II Eichler integrals of Eisenstein series � i ∞ ˜ [ f ( z ) − a (0)] ( z − τ ) k − 2 d z f ( τ ) = τ On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 10 / 33

Eichler integrals 1 d D = derivative 2 πi d τ h ∂ h = D − Maass raising operator 4 πy ∂ h ( F | h γ )=( ∂ h F ) | h +2 γ ∂ n h = ∂ h +2( n − 1) ◦ · · · ◦ ∂ h +2 ◦ ∂ h On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 11 / 33

Eichler integrals 1 d D = derivative 2 πi d τ h ∂ h = D − Maass raising operator 4 πy ∂ h ( F | h γ )=( ∂ h F ) | h +2 γ ∂ n h = ∂ h +2( n − 1) ◦ · · · ◦ ∂ h +2 ◦ ∂ h • By induction on n , following Lewis–Zagier 2001, n � j ∂ n D n − j � n + h − 1 � � − 1 h � n ! = ( n − j )! . j 4 πy j =0 On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 11 / 33

Eichler integrals 1 d D = derivative 2 πi d τ h ∂ h = D − Maass raising operator 4 πy ∂ h ( F | h γ )=( ∂ h F ) | h +2 γ ∂ n h = ∂ h +2( n − 1) ◦ · · · ◦ ∂ h +2 ◦ ∂ h • By induction on n , following Lewis–Zagier 2001, n � j ∂ n D n − j � n + h − 1 � � − 1 h � n ! = ( n − j )! . j 4 πy j =0 • In the special case n = 1 − h , with h = 2 − k , ∂ k − 1 2 − k = D k − 1 . On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 11 / 33

Eichler integrals THM For all sufficiently differentiable F and all γ ∈ SL 2 ( Z ) , Bol 1949 D k − 1 ( F | 2 − k γ ) = ( D k − 1 F ) | k γ. EG � aτ + b � � � aτ + b �� ( DF ) | 2 γ = ( cτ + d ) − 2 F ′ = D F k = 2 cτ + d cτ + d On a secant Dirichlet series and Eichler integrals of Eisenstein series Armin Straub 12 / 33