Trigonometric Dirichlet series and Eichler integrals Number Theory - PowerPoint PPT Presentation

Trigonometric Dirichlet series and Eichler integrals Number Theory and Experimental Mathematics Day Dalhousie University Armin Straub October 20, 2014 University of Illinois at Urbana–Champaign Based on joint work with : Bruce Berndt University of Illinois at Urbana–Champaign Trigonometric Dirichlet series and Eichler integrals Armin Straub 1 / 24

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 . Trigonometric Dirichlet series and Eichler integrals Armin Straub 2 / 24

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 Trigonometric Dirichlet series and Eichler integrals Armin Straub 2 / 24

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 . Trigonometric Dirichlet series and Eichler integrals Armin Straub 2 / 24

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 Trigonometric Dirichlet series and Eichler integrals Armin Straub 3 / 24

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 Trigonometric Dirichlet series and Eichler integrals Armin Straub 3 / 24

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. Trigonometric Dirichlet series and Eichler integrals Armin Straub 3 / 24

Secant zeta function: Convergence � sec( πnτ ) • ψ s ( τ ) = 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 / 1000 Re ψ 2 ( τ + εi ) with ε = 1 / 100 Trigonometric Dirichlet series and Eichler integrals Armin Straub 4 / 24

Secant zeta function: Convergence � sec( πnτ ) • ψ s ( τ ) = 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 / 1000 Re ψ 2 ( τ + εi ) with ε = 1 / 100 � sec( πnτ ) THM The series ψ s ( τ ) = converges absolutely if Lal´ ın– n s 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 Trigonometric Dirichlet series and Eichler integrals Armin Straub 4 / 24

Secant zeta function: Functional equation � sec( πnτ ) • Obviously, ψ s ( τ ) = 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 ) Trigonometric Dirichlet series and Eichler integrals Armin Straub 5 / 24

Secant zeta function: Functional equation � sec( πnτ ) • Obviously, ψ s ( τ ) = 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 ) → ∞ . Trigonometric Dirichlet series and Eichler integrals Armin Straub 5 / 24

Secant zeta function: Functional equation � sec( πnτ ) • Obviously, ψ s ( τ ) = 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 ) → ∞ . Trigonometric Dirichlet series and Eichler integrals Armin Straub 5 / 24

Secant zeta function: Functional equation � sec( πnτ ) • Obviously, ψ s ( τ ) = 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 ) 2 τ + 1 ψ 2 ( τ ) + π 2 τ (3 τ 2 + 4 τ + 2) EG � τ � 1 ψ 2 = 6(2 τ + 1) 2 2 τ + 1 Trigonometric Dirichlet series and Eichler integrals Armin Straub 5 / 24

Secant zeta function: Functional equation � sec( πnτ ) • Obviously, ψ s ( τ ) = 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 ) 2 τ + 1 ψ 2 ( τ ) + π 2 τ (3 τ 2 + 4 τ + 2) EG � τ � 1 ψ 2 = 6(2 τ + 1) 2 2 τ + 1 � 1 � � 1 � 2 0 • Hence, ψ 2 m transforms under T 2 = and R 2 = , 0 1 2 1 • Together, with − I , these two matrices generate Γ(2) . Trigonometric Dirichlet series and Eichler integrals Armin Straub 5 / 24

Secant zeta function: Special values For any positive rational r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 Trigonometric Dirichlet series and Eichler integrals Armin Straub 6 / 24

Secant zeta function: Special values For any positive rational r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 √ 2 is fixed by τ �→ 3 τ + 4 EG • 2 τ + 3 . Trigonometric Dirichlet series and Eichler integrals Armin Straub 6 / 24

Secant zeta function: Special values For any positive rational r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 √ 2 is fixed by τ �→ 3 τ + 4 EG • 2 τ + 3 . • We have the functional equation 2 τ + 3 ψ 2 ( τ ) − ( τ + 2)(3 τ 2 + 8 τ + 6) � 3 τ + 4 � 1 π 2 . ψ 2 = − 6(2 τ + 3) 2 2 τ + 3 Trigonometric Dirichlet series and Eichler integrals Armin Straub 6 / 24

Secant zeta function: Special values For any positive rational r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 √ 2 is fixed by τ �→ 3 τ + 4 EG • 2 τ + 3 . • We have the functional equation 2 τ + 3 ψ 2 ( τ ) − ( τ + 2)(3 τ 2 + 8 τ + 6) � 3 τ + 4 � 1 π 2 . ψ 2 = − 6(2 τ + 3) 2 2 τ + 3 √ • For τ = 2 this reduces to √ √ √ √ 2) + 2 2 − 2) π 2 . ψ 2 ( 2) = (2 2 − 3) ψ 2 ( 3( Trigonometric Dirichlet series and Eichler integrals Armin Straub 6 / 24

Secant zeta function: Special values For any positive rational r , THM LRR, BS ψ 2 m ( √ r ) ∈ Q · π 2 m . 2013 √ 2 is fixed by τ �→ 3 τ + 4 EG • 2 τ + 3 . • We have the functional equation 2 τ + 3 ψ 2 ( τ ) − ( τ + 2)(3 τ 2 + 8 τ + 6) � 3 τ + 4 � 1 π 2 . ψ 2 = − 6(2 τ + 3) 2 2 τ + 3 √ • For τ = 2 this reduces to √ √ √ √ 2) + 2 2 − 2) π 2 . ψ 2 ( 2) = (2 2 − 3) ψ 2 ( 3( √ 2) = − π 2 • Hence, ψ 2 ( 3 . Trigonometric Dirichlet series and Eichler integrals Armin Straub 6 / 24

Modular forms “ There’s a saying attributed to Eichler that there are five funda- mental operations of arithmetic: addition, subtraction, multipli- ” cation, division, and modular forms. Andrew Wiles (BBC Interview, “The Proof”, 1997) � a b DEF � Actions of γ = ∈ SL 2 ( Z ) : c d γ · τ = aτ + b • on τ ∈ H by cτ + d , ( f | k γ )( τ ) = ( cτ + d ) − k f ( γ · τ ) . • on f : H → C by Trigonometric Dirichlet series and Eichler integrals Armin Straub 7 / 24

Modular forms “ There’s a saying attributed to Eichler that there are five funda- mental operations of arithmetic: addition, subtraction, multipli- ” cation, division, and modular forms. Andrew Wiles (BBC Interview, “The Proof”, 1997) � a b DEF � Actions of γ = ∈ SL 2 ( Z ) : c d γ · τ = aτ + b • on τ ∈ H by cτ + d , ( f | k γ )( τ ) = ( cτ + d ) − k f ( γ · τ ) . • on f : H → C by A function f : H → C is a modular form of weight k if DEF • f | k γ = f for all γ ∈ Γ , Γ � SL 2 ( Z ) , • f is holomorphic (including at the cusps). Trigonometric Dirichlet series and Eichler integrals Armin Straub 7 / 24