HAL Id: hal-03195547
https://hal.archives-ouvertes.fr/hal-03195547
Submitted on 11 Apr 2021
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
RATIONALITY OF SECANT ZETA VALUES
Pierre Charollois, Matthew Greenberg
To cite this version:
Pierre Charollois, Matthew Greenberg. RATIONALITY OF SECANT ZETA VALUES. Annales mathématiques du Québec, Springer, 2014. �hal-03195547�
PIERRE CHAROLLOIS AND MATTHEW GREENBERG
Abstract. We use the Arakawa-Berndt theory of generalized η-functions to prove a con-jecture of Lal´ın, Rodrigue and Rogers concerning the algebraic nature of special values of the secant zeta function.
1. Introduction
The cotangent and secant zeta functions attached to an algebraic irrational number α are defined, for Re(s) large enough, by the Dirichlet series
ξ(α, s) = ∞ X n=1 cot(πnα) ns and ψ(α, s) = ∞ X n=1 sec(πnα) ns ,
respectively. The cotangent zeta function was introduced for s = 2k + 1, k ≥ 1 an integer, by Lerch [7], and for general s by Berndt [4] in the course of his study of generalized Dedekind sums. Lerch stated (without proof) the following functional equation valid for algebraic irrational α and sufficiently large k = k(α):
(1) ξ(α, 2k + 1) + α2kξ(1 α, 2k + 1) = (2π) 2k+1ϕ(α, 2k + 1), where ϕ(α, n) = n+1 X j=0 BjBn+1−j j!(n + 1 − j)!α j−1,
and Bk is the k-th Bernoulli number. Suppose α 6= ±1 is a unit in the quadratic field Q(√d) of discriminant d. Writing α = a+b
√ d
2 and defining ε = ±1 by a
2− b2d = ε, we have 1
α = −ε(α − a). Using the obvious identities ξ(−α, s) = −ξ(α, s) and ξ(α − 1, s) = ξ(α, s) together with (1), we deduce the following rationality result:
(2) ξ(α, 2k + 1) (2π)2k+1√d = 1 √ d ϕ(α, 2k + 1) 1 − εα2k ∈ Q.
Another proof of this result was given by Berndt [4, Theorem 5.2]. More generally, if α is an arbitrary real quadratic irrationality, then Lerch uses the continued fraction expansion of α to conclude that the left hand side of (2) belongs to Q(α) – see [7, (3)].
Lal´ın, Rodrigue and Rogers conjecture an analogous result for the secant zeta function:
Date: February 20, 2014.
2010 Mathematics Subject Classification. 11F11.
PC’s research is partially supported by the grant REGULATEURS “ANR-12-BS01-0002”. MG’s research is supported by NSERC of Canada.
2 PIERRE CHAROLLOIS AND MATTHEW GREENBERG
Conjecture 1.1 ([6, Conjecture 1]). Suppose d and k are positive integers such that d is not a square. Then
ψ(√d, 2k) π2k ∈ Q.
They prove a functional equation for ψ(α, s) analogous to (1) and use it to deduce many instances of Conjecture 1.1 as above. In §2, we show that their functional equation is actually sufficient to prove in general that ψ(α, 2k) ∈ π2kQ(α) for all real quadratic irrationalities α, not merely those of the form α =√d.
In §3, we relate ψ(α, s) to the generalized η-functions studied by Arakawa [1], fascinating objects in their own right. Using this relationship, we leverage Arakawa’s results to give an explicit formula for ψ(α, 2k) for real quadratic irrationalities α, yielding another proof of Conjecture 1.1.
2. The Lal´ın-Rodrigue-Rogers functional equation
One can prove Conjecture 1.1 following Lerch’s approach for the cotangent zeta function. Most of this argument was given in [6]; we complete their thought. Let A, B ∈ PSL2(Z) be defined by A =1 2 0 1 , B =1 0 2 1 .
The following functional equations are established in [6, (4.1), (4.2)]: ψ(Aα, 2k) = ψ(α, 2k), ψ(Bα, 2k) = (2α + 1)1−2kψ(α, 2k) − π 2k (2k)! 2k X m=0 (2m−1− 1)BmE2k−m 2k m (α + 1)2k−m(2α + 1)m−2k− (2α + 1)1−2k.
Here, Bn is the n-th Bernoulli number and En is the n-th Euler number. It follows that if C ∈ hA, Bi then there is a Q(α)-linear relation between ψ(Cα, 2k), ψ(α, 2k) and π2k. Thus, if there is a matrix C ∈ hA, Bi such that Cα = α then ψ(α, 2k) ∈ π2kQ(α). In [6, §4], several families of examples of such pairs (α, C) are given and the associated linear relations are worked out explicitly. We merely point out that if α is any real quadratic irrationality, then there is always a C ∈ hA, Bi, C 6= 1 such that Cα = α.
To see this, let α be a real quadratic irrationality and consider the lattice L = Z + Zα ⊂ Q(α). Let O be the order of L:
O = {x ∈ Q(α) : xL ⊂ L}.
Then O is an order in Q(α). Let u ∈ O. Writing u · α = aα + b and u · 1 = cα + d with a, b, c, d ∈ Z, we have uα 1 =a b c d α 1 . Write j(u) for the matrix (a b
c d) associated to u above. Then j : O → M2(Z) is a ring homomorphism. Since (α1) is an eigenvector of j(u) for all u ∈ O, we have j(u)α := aα+bcα+d = α when u 6= 0. By Dirichlet’s unit theorem, the group O∗+ of totally positive units in O is free of rank 1; write O∗+ = hγi. By [9, p. 84], hA, Bi is the principal congruence subgroup
Γ(2) ⊂ PSL2(Z), this inclusion having index 6. Therefore, C := j(γ6) satisfies C 6= 1, C ∈ Γ(2) and Cα = α.
3. Generalized η-functions and secant zeta values
Arakawa [2] gave another proof of (2) by relating ξ(α, s) to generalized η-functions, the theory of which he developed in [1]. In turn, Arakawa’s work has its foundations in papers of Lewittes [8] and Berndt [3]. We show that Arakawa’s method can also be used to analyze the secant zeta function.
For x ∈ R, define hxi (resp., {x}) by
0 < hxi ≤ 1 (resp. 0 ≤ {x} < 1) and x − hxi ∈ Z (resp., x − {x} ∈ Z). We set e(z) = e2πiz.
Following [1], let p, q ∈ R and define
η(α, s, p, q) = ∞ X n=1 ns−1e(n(pα + q)) 1 − e(nα) H(α, s, p, q) = η(α, s, hpi, q) + e s2 η(α, s, h−pi, −q). Theorem 3.1 ([1, Lemma 1 and Theorem 2]). Suppose
α ∈ R ∩ ¯Q and α /∈ Q.
Then η(α, s, p, q) is absolutely convergent for <(s) < 0. If, in addition, [Q(α) : Q] = 2 and p, q ∈ Q
then H(α, s, p, q) has analytic continuation to C − {0}, and the singularity at s = 0 is at worst a simple pole.
Remark 3.2. The convergence of η(α, s, p, q) relies on the Thue-Siegel-Roth theorem in much the same way that the convergence of ψ(α, s) does – see [6, Theorem 1].
An elementary computation yields
ξ(α, s) = −2i H(α, 1 − s, 1, 0) 1 + e(s2) + 1 2ζ(s) ,
so rationality statements for H(α, 1 − s, 1, 0) for even integral s translate to rationality statements for ξ(α, s) at odd integral s. In contrast, ψ(α, s) does not seem to have a simple expression in terms of H(α, s, p, q). Crucially, however, we still have a relation between certain special values of H and ψ relying on the relation 34 = −14 + 1 :
ψ α2, 1 − s = ∞ X n=1 ns−1 2 e(nα4 ) + e(−nα4 ) = 2 ∞ X n=1 ns−1 e( nα 4 ) 1 − e(nα) − 2 ∞ X n=1 ns−1 e( 3nα 4 ) 1 − e(nα) = 2η(α, s, h14i, 0) − 2η(α, s, h−1 4i, 0). If s = 1 − 2k, then e(s
2) = −1 and we conclude that
4 PIERRE CHAROLLOIS AND MATTHEW GREENBERG
For the rest of the paper, suppose that α is a real quadratic irrationality. Formulas of Berndt and Arakawa allow us to evaluate H(α, 1 − 2k,14, 0) rather explicitly. Let
V =a b c d
∈ SL2(Z)
be a matrix such that
c > 0 and β := cα + d > 0. Set
(p0, q0) = (p, q)V and % = {q0}c − {p0}d.
Theorem 3.3 ([1, Theorem 1 and Eq. (1.19)]). Suppose that p and p0 are not in Z. Then the following transformation formula holds:
β−sH(V α, s, p, q) = H(α, s, p0, q0) + (2π)−se(−s
4)L(α, s, p 0
, q0, c, d), where L(α, s, p0, q0, c, d) is as in (2) of [3]. If s = −m is a negative integer, then
L(α, −m, p0, q0, c, d) = 2πi c X j=1 m+2 X `=0 b` j − {p0} c bm+2−` jd + % c (−β)`−1.
Here, b` is the (normalized) `-th Bernoulli polynomial defined by the generating series ueux
eu − 1 = X
`≥0
b`(x)u`.
By [1, Lemma 4], for any rational numbers p and q there is a totally positive unit β of Q(α) and a matrix V ∈ SL2(Z) such that
c > 0, (p0, q0) := (p, q)V ≡ (p, q) (mod Z2), and V α 1 = βα 1 .
The last condition implies that β = cα + d, consistently with the notation introduced above. Suppose p /∈ Z . Then p0 ∈ Z, too, as p ≡ p/ 0 (mod Z). Applying Theorem 3.3, observing that H and L only depend on the class of (p, q) ≡ (p0, q0) modulo Z2, and rearranging terms, we get
(β−s− 1)H(α, s, p, q) = (2π)−se(−s4)L(α, s, p, q, c, d). By the second part of Theorem 3.3, if s = 1 − 2k then
(4) H(α, 1 − 2k, p, q) π2k = 22k(−1)k (β2k−1− 1) c X j=1 2k+1 X `=0 b` j − {p} c b2k+1−` jd + % c (−β)`−1.
Setting (p, q) = (14, 0) and using (3), this formula specializes to
(5) ψ( α 2, 2k) π2k = 22k+1(−1)k (β2k−1− 1) c X j=1 2k+1 X `=0 b` j − 1 4 c b2k+1−` d(j − 1 4) c (−β)`−1. We conclude :
Theorem 3.4. Suppose α is a real quadratic irrationality and k is a positive integer. Then ψ(α, 2k)
π2k ∈ Q(α).
Moreover, if x 7→ x0 is the nontrivial automorphism of Q(α), then ψ(α, 2k) π2k 0 = ψ(α 0, 2k) π2k .
Conjecture 1.1 follows from Theorem 3.4 and the evenness of the secant function.
References
[1] Tsuneo Arakawa, Generalized eta-functions and certain ray class invariants of real quadratic fields, Math. Ann. 260 (1982), 475-494.
[2] Tsuneo Arakawa, Dirichlet series P∞
n=1cot(πnα)/n
s, Dedekind sums, and Hecke L-functions for real
quadratic fields, Comment. Math. Univ. St. Paul. 37 (1988), 209-235.
[3] Bruce C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums, Trans. AMS 178 (1973), 495-508.
[4] Bruce C. Berndt, Dedekind sums and a paper of G. H. Hardy, J. London Math. Soc. (2) 13 (1976), 129-137.
[5] Bruce C. Berndt, Armin Straub, On a secant Dirichlet series and Eichler integrals of Eisenstein series. preprint (2013), available at http://arminstraub.com/downloads/pub/secantsums.pdf.
[6] Matilde Lal´ın, Francis Rodrigue and Matthew Rogers, Secant zeta functions, J. Math. Anal. Appl. 409 (2014), no. 1, 197 – 204.
[7] Mathias Lerch, Sur une s´erie analogue aux fonctions modulaires, C. R. Acad. Sci. Paris, 138 (1904), 952-954.
[8] Joseph Lewittes, Analytic continuation of Eisenstein series, Trans. AMS 177 (1972), 469-490.
[9] Bruno Schoeneberg, Elliptic modular functions (Die Grundlehren der Math. Wiss. 203), Springer-Verlag (1974).
E-mail address: charollois@math.jussieu.fr
Institut de math´ematiques de Jussieu, Universit´e Paris 6, 4 Place Jussieu, 75005 Paris, France
E-mail address: mgreenbe@ucalgary.ca
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
