On a secant Dirichlet series and Eichler integrals of Eisenstein series

On a secant Dirichlet series and Eichler integrals of Eisenstein series Oberseminar Zahlentheorie Universität zu Köln University of Illinois at Urbana Champaign November 12, 2013 & Max-Planck-Institut für Mathematik, Bonn Based on joint work with: Bruce Berndt University of Illinois at Urbana Champaign 1 / 33

PART I A secant Dirichlet series ψ s (τ) = n=1 sec(πnτ) n s 2 / 33

Secant zeta function Laĺın, Rodrigue and Rogers introduce and study ψ s (τ) = n=1 sec(πnτ) n s. Clearly, ψ s (0) = ζ(s). In particular, ψ 2 (0) = π2 6. EG LRR 13 ψ 2 ( 2) = π2 3, ψ 2( 6) = 2π2 3 3 / 33

Secant zeta function Laĺın, Rodrigue and Rogers introduce and study ψ s (τ) = n=1 sec(πnτ) n s. Clearly, ψ s (0) = ζ(s). In particular, ψ 2 (0) = π2 6. EG LRR 13 ψ 2 ( 2) = π2 3, ψ 2( 6) = 2π2 3 CONJ LRR 13 For positive integers m, r, ψ 2m ( r) Q π 2m. 3 / 33

Secant zeta function: Motivation Euler s identity: n=1 1 n 2m = 1 2 (2πi)2m B 2m (2m)! Half of the Clausen and Glaisher functions reduce, e.g., n=1 cos(nτ) n 2m = poly m(τ), poly 1 (τ) = τ 2 4 πτ 2 + π2 6. Ramanujan investigated trigonometric Dirichlet series of similar type. From his first letter to Hardy: n=1 coth(πn) n 7 = 19π7 56700 In fact, this was already included in a general formula by Lerch. 4 / 33

Secant zeta function: Convergence ψ s (τ) = sec(πnτ) n s has singularity at rationals with even denominator 6 10 4 5 2 2 0.2 0.4 0.6 0.8 5 0.2 0.4 0.6 0.8 4 ψ 2 (τ) truncated to 4 and 8 terms 10 Re ψ 2 (τ + εi) with ε = 1/1000 THM Laĺın Rodrigue Rogers 2013 The series ψ s (τ) = sec(πnτ) n s 1 τ = p/q with q odd and s > 1, 2 τ is algebraic irrational and s 2. converges absolutely if Proof uses Thue Siegel Roth, as well as a result of Worley when s = 2 and τ is irrational 5 / 33

Secant zeta function: Functional equation Obviously, ψ s (τ) = sec(πnτ) n s THM LRR, BS 2013 (1 + τ) 2m 1 ψ 2m ( τ 1 + τ = π 2m rat(τ) satisfies ψ s (τ + 2) = ψ s (τ). ) (1 τ) 2m 1 ψ 2m ( ) τ 1 τ 6 / 33

Secant zeta function: Functional equation Obviously, ψ s (τ) = sec(πnτ) n s THM LRR, BS 2013 (1 + τ) 2m 1 ψ 2m ( τ 1 + τ = π 2m rat(τ) satisfies ψ s (τ + 2) = ψ s (τ). ) (1 τ) 2m 1 ψ 2m ( ) τ 1 τ proof Collect residues of the integral I C = 1 sin (πτz) dz 2πi sin(π(1 + τ)z) sin(π(1 τ)z) z s+1. C C are appropriate circles around the origin such that I C 0 as radius(c). 6 / 33

Secant zeta function: Functional equation Obviously, ψ s (τ) = sec(πnτ) n s THM LRR, BS 2013 (1 + τ) 2m 1 ψ 2m ( τ 1 + τ satisfies ψ s (τ + 2) = ψ s (τ). ) (1 τ) 2m 1 ψ 2m ( = π 2m [z 2m 1 sin(τ z) ] sin((1 τ)z) sin((1 + τ)z) ) τ 1 τ proof Collect residues of the integral I C = 1 sin (πτz) dz 2πi sin(π(1 + τ)z) sin(π(1 τ)z) z s+1. C C are appropriate circles around the origin such that I C 0 as radius(c). 6 / 33

Secant zeta function: Functional equation Obviously, ψ s (τ) = sec(πnτ) n s THM LRR, BS 2013 DEF slash operator (1 + τ) 2m 1 ψ 2m ( τ 1 + τ satisfies ψ s (τ + 2) = ψ s (τ). ) (1 τ) 2m 1 ψ 2m ( = π 2m [z 2m 1 sin(τ z) ] sin((1 τ)z) sin((1 + τ)z) F k ( a b c d ) (τ) = (cτ + d) k F ( ) aτ + b cτ + d ) τ 1 τ 6 / 33

Secant zeta function: Functional equation Obviously, ψ s (τ) = sec(πnτ) n s THM LRR, BS 2013 DEF slash operator In terms of (1 + τ) 2m 1 ψ 2m ( τ 1 + τ satisfies ψ s (τ + 2) = ψ s (τ). ) (1 τ) 2m 1 ψ 2m ( = π 2m [z 2m 1 sin(τ z) ] sin((1 τ)z) sin((1 + τ)z) F k ( a b c d ) (τ) = (cτ + d) k F T = ( ) 1 1, S = 0 1 the functional equations become ψ 2m 1 2m (T 2 1) = 0, ( ) aτ + b cτ + d ( ) 0 1, R = 1 0 ψ 2m 1 2m (R 2 1) = π 2m rat(τ). ) τ 1 τ ( ) 1 0, 1 1 6 / 33

Secant zeta function: Functional equation The matrices T 2 = together with I, generate ( ) 1 2, R 2 = 0 1 ( ) 1 0, 2 1 Γ(2) = {γ SL 2 (Z) : γ I (mod 2)}. COR For any γ Γ(2), ψ 2m 1 2m (γ 1) = π 2m rat(τ). 7 / 33

Secant zeta function: Special values THM LRR, BS 2013 For positive integers m, r, ψ 2m ( r) Q π 2m. proof Note that ( X ry Y X ) r = r. As shown by Lagrange, there are X and Y which solve Pell s equation X 2 ry 2 = 1. Since ( ) 2 X ry γ = = Y X ( X 2 + ry 2 ) 2rXY 2XY X 2 + ry 2 Γ(2), the claim follows from the evenness of ψ 2m and ψ 2m 1 2m (γ 1) = π 2m rat(τ). 8 / 33

Secant zeta function: Special values EG For integers κ, µ, ψ 2 (κ + ψ 4 (κ + κ κ ( 1 µ + κ ) ) = π2 6 ( 1 µ + κ ) ) = π4 90 ( 1 + 3κ ), 2µ ( 1 + 5κ 2µ 5κ2 (16µ 2 15) 8µ 2 (4κµ + 3) ). 9 / 33

PART II Eichler integrals of Eisenstein series f(τ) = i τ [f(z) a(0)] (z τ) k 2 dz 10 / 33

Eichler integrals D = 1 d 2πi dτ h = D h 4πy derivative Maass raising operator h (F h γ)=( h F ) h+2 γ n h = h+2(n 1) h+2 h 11 / 33

Eichler integrals D = 1 d 2πi dτ h = D h 4πy derivative Maass raising operator 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, h n n ( ) ( n + h 1 n! = 1 ) j D n j j 4πy (n j)!. j=0 11 / 33

Eichler integrals THM Bol 1949 For all sufficiently differentiable F and all γ SL 2 (Z), D k 1 (F 2 k γ) = (D k 1 F ) k γ. EG k = 2 ( ) [ aτ + b (DF ) 2 γ = (cτ + d) 2 F = D F cτ + d ( )] aτ + b cτ + d 12 / 33

Eichler integrals THM Bol 1949 For all sufficiently differentiable F and all γ SL 2 (Z), D k 1 (F 2 k γ) = (D k 1 F ) k γ. EG k = 2 ( ) [ aτ + b (DF ) 2 γ = (cτ + d) 2 F = D F cτ + d ( )] aτ + b cτ + d F is an Eichler integral if D k 1 F is modular of weight k. Then D k 1 (F 2 k γ) = D k 1 F, and hence F 2 k (γ 1) = poly(τ), deg poly k 2. poly(τ) is a period polynomial of the modular form. 12 / 33

Eichler integrals For modular f(τ) = a(n)q n, weight k, define the Eichler integral f(τ) = i τ [f(z) a(0)] (z τ) k 2 dz = ( 1)k Γ(k 1) (2πi) k 1 n=1 a(n) n k 1 qn. If a(0) = 0, f is an Eichler integral in the strict sense of the previous slide. EG For cusp forms f of level 1, the period polynomial ρ f (X) is f 2 k (S 1) = i 0 f(z)(z X) k 2 dz k 1 ( ) k 2 Γ(s) = ( 1) k s 1 (2πi) s L(f, s)xk s 1. s=1 13 / 33

The period polynomials Let U = T S = ( ) 1 1 1 0, and define { W k 2 = p C[X] : p 2 k (1 + S) = p 2 k (1 + U + U 2 ) = 0 deg p=k 2 }. 14 / 33

The period polynomials Let U = T S = ( ) 1 1 1 0, and define { W k 2 = p C[X] : p 2 k (1 + S) = p 2 k (1 + U + U 2 ) = 0 deg p=k 2 Denote with p the odd part of p. THM Eichler Shimura The space of (level 1) cusp forms S k is isomorphic to W k 2 via f ρ f (X). }. Similarly, W k 2 is isomorphic to S k M k. 14 / 33

The period polynomials in higher level Let Γ be of finite index in Γ 1 = SL 2 (Z). Let V k 2 be the polynomials of degree at most k 2. DEF Paşol Popa 2013 The multiple period polynomial of f is ρ f (A)(X) = ρ f : Γ\Γ 1 V k 2, i 0 [f A(z) a 0 (f A)] (z X) k 2 dz. γ Γ 1 acts on p : Γ\Γ 1 V k 2 by p γ(a) = p(aγ 1 ) 2 k γ. Paşol and Popa extend Eichler Shimura isomorphism to this settinsetting. 15 / 33

Eichler integrals of Eisenstein series For the Eisenstein series G 2k, G 2k (τ) = 2ζ(2k) + 2 (2πi)2k Γ(2k) G 2k (τ) = 4πi 2k 1 n=1 σ 2k 1 (n)q n, n=1 σ 2k 1 (n) n 2k 1 q n. n 1 2k q n 1 q n n 2k 1 q n 1 q n 16 / 33

Eichler integrals of Eisenstein series For the Eisenstein series G 2k, G 2k (τ) = 2ζ(2k) + 2 (2πi)2k Γ(2k) G 2k (τ) = 4πi 2k 1 n=1 σ 2k 1 (n)q n, n=1 σ 2k 1 (n) n 2k 1 q n. n 1 2k q n 1 q n The period polynomial G 2k 2 2k (S 1) is given by (2πi) 2k 2k 1 [ k s=0 n 2k 1 q n 1 q n ] B 2s B 2k 2s ζ(2k 1) (2s)! (2k 2s)! X2s 1 + (2πi) 2k 1 (X2k 2 1). 16 / 33

Ramanujan s formula THM Ramanujan, Grosswald For α, β > 0 such that αβ = π 2 and m Z, α m { ζ(2m + 1) 2 2 2m m+1 + n=1 n 2m 1 e 2αn 1 } = ( β) m { ζ(2m + 1) 2 ( 1) n B 2n B 2m 2n+2 (2n)! (2m 2n + 2)! αm n+1 β n. n=0 + n=1 n 2m 1 e 2βn 1 } In terms of ξ s (τ) = n=1 Ramanujan s formula takes the form cot(πnτ) n s, 1 e x 1 = 1 2 cot( x 2 ) 1 2 k ξ 2k 1 2 2k (S 1) = ( 1) k (2π) 2k 1 B 2s B 2k 2s (2s)! (2k 2s)! τ 2s 1. s=0 17 / 33

Secant zeta function cot(πnτ) n 2k 1 is an Eichler integral of the Eisenstein series G 2k. EG cot(πτ) = 1 π j Z 1 τ + j lim N N j= N 18 / 33

Secant zeta function cot(πnτ) n 2k 1 is an Eichler integral of the Eisenstein series G 2k. EG cot(πτ) = 1 π j Z 1 τ + j lim N N j= N sec(πnτ) n 2k EG is an Eichler integral of an Eisenstein series with character. sec ( πτ ) = 2 χ 4 (j) 2 π τ + j j Z m,n Z χ 4 (n) is an Eisenstein series of weight 2k + 1. (mτ + n) 2k+1 18 / 33

Eisenstein series More generally, we have the Eisenstein series E k (τ; χ, ψ) = m,n Z χ(m)ψ(n) (mτ + n) k, where χ and ψ are Dirichlet characters modulo L and M. We assume χ( 1)ψ( 1) = ( 1) k. Otherwise, E k (τ; χ, ψ) = 0. PROP Modular transformations: γ = ( ) a Mb Lc d SL2 (Z) E k (τ; χ, ψ) k γ = χ(d) ψ(d)e k (τ; χ, ψ) E k (τ; χ, ψ) k S = χ( 1)E k (τ; ψ, χ) PROP If ψ is primitive, the L-function of E(τ) = E k (τ; χ, ψ) is L(E, s) = const M s L(χ, s)l( ψ, 1 k + s). 19 / 33

Generalized Bernoulli numbers EG Euler ζ(2n) = 1 2 (2πi)2n B 2n (2n)! For integer n > 0 and primitive χ with χ( 1) = ( 1) n, (χ of conductor L and Gauss sum G(χ)) n 1 G(χ) L(n, χ) = ( 1) 2 L(1 n, χ) = B n,χ /n. ( ) 2πi n B n, χ, L n! The generalized Bernoulli numbers have generating function n=0 B n,χ x n n! = L a=1 χ(a)xe ax e Lx 1. 20 / 33

Period polynomials of Eisenstein series THM Berndt-S 2013 For k 3 and primitive χ 1, ψ 1, Ẽ k (X; χ, ψ) ψ( 1)X k 2 Ẽ k ( 1/X; ψ, χ) k B k s, χ B s, ψ = const (k s)!l k s s!m s Xs 1. s=0 const = χ( 1)G (χ) G(ψ) (2πi)k k 1 If χ or ψ are principal, then we need to add to the RHS: 2ψ( 1) [ ] k 1 πi δ χ=1 L(k 1, ψ)x k 2 δ ψ=1 L(k 1, χ) Recall that we assume χ( 1)ψ( 1) = ( 1) k. 21 / 33

Period polynomials of Eisenstein series COR Berndt-S 2013 For k 3, primitive χ, ψ 1, and n such that L n, Ẽ k (X; χ, ψ) 2 k (1 R n ) k = const s=0 R n = ( 1 0 n 1 ) B k s, χ (k s)!l k s B s, ψ s!m s Xs 1 2 k (1 R n ). Note that X s 1 2 k (1 R n ) = X s 1 (1 (nx + 1) k 1 s ). 22 / 33

Application: Grosswald-type formula for Dirichlet L-values THM Berndt-S 2013 For α H, such that R k (α; χ, 1) = 0 and α k 2 1, (k 3, χ primitive, χ( 1) = ( 1) k ) [ Ẽ k ( α 1 k 1 L(k 1, χ) = 2πi(1 α k 2 ) 2 [ χ(n) = 1 α k 2 n k 1 n=1 ) L ; χ, 1 ( )] 1 1/α α k 2 Ẽ k ; χ, 1 L 1 1 e 2πin(1 α)/l α k 2 1 e 2πin(1/α 1)/L ]. THM Gun Murty Rath 2011 As β H, β 2k 2 1, ranges over algebraic numbers, the values 1 [Ẽ2k (β; 1, 1) β 2k 2 Ẽ 2k ( 1/β; 1, 1)] π contain at most one algebraic number. 23 / 33

PART III Unimodularity of period polynomials R k (X) = k s=0 B s s! B k s (k s)! Xs 1 24 / 33

Unimodular polynomials DEF p(x) is unimodular if all its zeros have absolute value 1. Kronecker: if p(x) Z[x] is monic and unimodular, hence Mahler measure 1, then all of its roots are roots of unity. EG Lehmer EG x 10 + z 9 z 7 z 6 z 5 z 4 z 3 + z + 1 has only the two real roots 0.850, 1.176 off the unit circle. Lehmer s conjecture: 1.176... is the smallest Mahler measure (greater than 1) x 2 + 6 5 x + 1 = ( x + 3+4i 5 ) ( ) x + 3 4i 5 25 / 33

Unimodular polynomials DEF p(x) is unimodular if all its zeros have absolute value 1. Kronecker: if p(x) Z[x] is monic and unimodular, hence Mahler measure 1, then all of its roots are roots of unity. EG Lehmer EG THM Cohn 1922 x 10 + z 9 z 7 z 6 z 5 z 4 z 3 + z + 1 has only the two real roots 0.850, 1.176 off the unit circle. Lehmer s conjecture: 1.176... is the smallest Mahler measure (greater than 1) x 2 + 6 5 x + 1 = ( x + 3+4i 5 P (x) is unimodular if and only if ) ( ) x + 3 4i 5 P (x) = a 0 + a 1 x +... + a n x n is self-inversive, i.e. a k = εa n k for some ε = 1, and P (x) has all its roots within the unit circle. 25 / 33

Ramanujan polynomials Following Gun Murty Rath, the Ramanujan polynomials are k X Bs Bk s Rk (X) = X s 1. s! (k s)! s=0 THM MurtySmythWang 11 THM Lalı n-smyth 13 All nonreal zeros of Rk (X) lie on the unit circle. For k > 2, R2k (X) has exactly four real roots which approach ±2±1. R2k (X) + ζ(2k 1) 2k 2 (X 1) is unimodular. (2πi)2k 1 1.0 1.0 0.5 0.5 0.0 0.0-0.5-0.5 full R20 (X) R20 (X) -1.0-1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0 26 / 33

Ramanujan polynomials 1.0 1.0 0.5 0.5 0.0 0.0-0.5-0.5-1.0-1.0-1.0-0.5 0.0 0.5 1.0 R20 (X) -1.0-0.5 0.0 0.5 1.0 full (X) R20 27 / 33

Generalized Ramanujan polynomials Consider the following generalized Ramanujan polynomials: R k (X) = R k (X; χ, ψ) = k s=0 k s=0 B s s! B s,χ s! B k s (k s)! Xs 1 B k s,ψ (k s)! ( ) X 1 k s 1 (1 X s 1 ) M PROP Berndt-S 2013 For k > 1, R 2k (X; 1, 1) = R 2k (X). R k (X; χ, ψ) is self-inversive. 28 / 33

Generalized Ramanujan polynomials EG R k (X; χ, 1) For χ real, apparently unimodular unless: χ = 1: R 2k (X; 1, 1) has real roots approaching ±2 ±1 χ = 3 : R 2k+1 (X; 3, 1) has real roots approaching 2 ±1 EG Apparently: R k (X; 1, ψ) unimodular for ψ one of 3, 4, 5+, 8±, 11, 12+, 13+, 19, 21+, 24+,... all nonreal roots on the unit circle if ψ is one of 1+, 7, 15, 17+, 20, 23, 24,... four nonreal zeros off the unit circle if ψ is one of 35, 59, 83, 131, 155, 179,... 29 / 33

Generalized Ramanujan polynomials A second kind of generalized Ramanujan polynomials: R k (X) = S k (X; χ, ψ) = k s=0 k s=0 B s s! Obviously, S k (X; 1, 1) = R k (X). B s,χ s! B k s (k s)! Xs 1 B k s,ψ (k s)! ( ) LX k s 1 M 30 / 33

Generalized Ramanujan polynomials A second kind of generalized Ramanujan polynomials: R k (X) = S k (X; χ, ψ) = k s=0 k s=0 B s s! Obviously, S k (X; 1, 1) = R k (X). CONJ B s,χ s! B k s (k s)! Xs 1 B k s,ψ (k s)! ( ) LX k s 1 M If χ is nonprincipal real, then S k (X; χ, χ) is unimodular (up to trivial zero roots). 30 / 33

Generalized Ramanujan polynomials 1.0 1.0 0.5 0.5 0.0 0.0-0.5-0.5-1.0-1.0-1.0-0.5 0.0 0.5 1.0 R19 (X; 1, χ 4 ) -1.0-0.5 0.0 0.5 1.0 S20 (X; χ 4, χ 4 ) 31 / 33

Unimodularity of period polynomials Both kinds of generalized Ramanujan polynomials are, essentially, period polynomials: χ, ψ primitive, nonprincipal S k (X; χ, ψ) = const [Ẽk (X; χ, ψ) ψ( 1)X k 2 Ẽ k ( 1/X; ψ, ] χ) R k (LX + 1; χ, ψ) = S k (X; χ, ψ) 2 k (1 R L ) THM Conrey- Farmer- Imamoglu 2012 THM El-Guindy Raji 2013 = const Ẽk(X; χ, ψ) 2 k (1 R L ) For any Hecke cusp form (for SL 2 (Z)), the odd part of its period polynomial has trivial zeros at 0, ±2, ± 1 2, and all remaining zeros lie on the unit circle. For any Hecke eigenform (for SL 2 (Z)), the full period polynomial has all zeros on the unit circle. 32 / 33

THANK YOU! Slides for this talk will be available from my website: http://arminstraub.com/talks B. Berndt, A. Straub Preprint, 2013 33 / 33