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
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