RATIONALITY OF SECANT ZETA VALUES

Transcription

1 RATIONALITY OF SECANT ZETA VALUES PIERRE CHAROLLOIS AND MATTHEW GREENBERG Abstrat We use the Arakawa-Berndt theory of generalized η-funtions to prove a onjeture of Lalìn, Rodrigue and Rogers onerning the algebrai nature of speial values of the seant zeta funtion Introdution The otangent and seant zeta funtions attahed to an algebrai irrational number α are defined, for Re(s) large enough, by the Dirihlet series ot(πnα) se(πnα) ξ(α, s) = and ψ(α, s) =, n s n s respetively The otangent zeta funtion was introdued for s = 2k +, k an integer, by Lerh [6], and for general s by Berndt [] in the ourse of his study of generalized Dedekind sums Lerh stated (without proof) the following funtional equation valid for algebrai irrational α and suffiiently large k = k(α): ( ξ(α, 2k + + α 2k ξ( α, 2k + = (2π)2k+ ϕ(α, 2k +, where ϕ(α, n) = n+ j=0 B j B n+ j j!(n + j)! αj, and B k is the k-th Bernoulli number Suppose α ± is a unit in the quadrati field Q( d) of disriminant d Writing α = a+b d and defining ε = ± by a 2 b 2 d = ε, we have 2 = ε(α a) Using the obvious identities ξ( α, s) = ξ(α, s) and ξ(α, s) = ξ(α, s) α together with (, we dedue the following rationality result: (2) ξ(α, 2k + (2π) 2k+ d = ϕ(α, 2k + Q d εα 2k Another proof of this result was given by Berndt [, Theorem 52] More generally, if α is an arbitrary real quadrati irrationality, then Lerh uses the ontinued fration expansion of α to onlude that the left hand side of (2) belongs to Q(α) see [6, (3)] Lalìn, Rodrigue and Rogers onjeture an analogous result for the seant zeta funtion: Date: September 8, Mathematis Subjet Classifiation F PC s researh is partially supported by the grant REGULATEURS ANR-2-BS MG s researh is supported by NSERC of Canada

2 2 PIERRE CHAROLLOIS AND MATTHEW GREENBERG Conjeture ([5, Conjeture ]) Suppose d and k are positive integers suh that d is not a square Then ψ( d, 2k) π 2k Q They prove a funtional equation for ψ(α, s) analogous to ( and use it do dedue many instanes of Conjeture as above In 2, we show that their funtional equation is atually suffiient to prove in general that ψ(α, 2k) π 2k Q(α) for all real quadrati irrationalities α, not merely those of the form α = d In 3, we relate ψ(α, s) to the generalized η-funtions studied by Arakawa [], fasinating objets in their own right Using this relationship, we leverage Arakawa s results to give an expliit formula for ψ(α, 2k) for real quadrati irrationalities α, yielding another proof of Conjeture 2 The Lalìn-Rodrigue-Rogers funtional equation One an prove Conjeture following Lerh s approah for the otangent zeta funtion Most of this argument was given in [5]; we omplete their thought Let A, B PSL 2 (Z) be defined by A = 2, B = The following funtional equations are established in [5, (, (2)]: ψ(aα, 2k) = ψ(α, 2k), ψ(bα, 2k) = (2α + 2k ψ(α, 2k) π2k (2k)! 2k m=0 (2 m B m E 2k m ( 2k m )(α + 2k m ((2α + m 2k (2α + 2k ) Here, B n is the n-th Bernoulli number and E n is the n-th Euler number It follows that if C A, B then there is a Q(α)-linear relation between ψ(cα, 2k), ψ(α, 2k) and π 2k Thus, if there is a matrix C A, B suh that Cα = α then ψ(α, 2k) π 2k Q(α) In [5, ], several families of examples of suh pairs (α, C) are given and the assoiated linear relations are worked out expliitly We merely point out that if α is any real quadrati irrationality, then there is always a C A, B, C suh that Cα = α To see this, let α be a real quadrati irrationality and onsider the lattie 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) Writing u α = aα + b and u = α + d with a, b,, d Z, we have ( ( α a b α u = d) Write j(u) for the matrix ( a d b ) assoiated to u above Then j : O M 2(Z) is a ring homomorphism Sine ( α ) is an eigenvetor of j(u) for all u O, we have j(u)α = α when u 0 By Dirihlet s unit theorem, the group O+ of totally positive units in O is free of rank ; write O+ = γ By [8, p 8], A, B is the prinipal ongruene subgroup

3 RATIONALITY OF SECANT ZETA VALUES 3 Γ(2) PSL 2 (Z), this inlusion having index 6 Therefore, C := j(γ 6 ) satisfies C, C Γ(2) and Cα = α 3 Generalized η-funtions and seant zeta values Arakawa [2] gave another proof of (2) by relating ξ(α, s) to generalized η-funtions, the theory of whih he developed in [] In turn, Arakawa s work has its foundations in papers of Lewittes [7] and Berndt [3] We show that Arakawa s method an also be used to analyze the seant zeta funtion For x R, define x (resp, {x}) by 0 < x (resp 0 {x} < and x x Z (resp, x {x} Z) We set e(z) = e 2πiz Following [], let p, q R and define s e(n(pα + q)) η(α, s, p, q) = n e(nα) H(α, s, p, q) = η(α, s, p, q) + e ( s 2) η(α, s, p, q) Theorem 3 ([, Lemma and Theorem 2]) Suppose α R Q and α / Q Then η(α, s, p, q) is absolutely onvergent for R(s) < 0 If, in addition, [Q(α) : Q] = 2 and p, q Q then H(α, s, p, q) has analyti ontinuation to C {0}, and the singularity at s = 0 is at worst a simple pole Remark 32 The onvergene of η(α, s, p, q) relies on the Thue-Siegel-Roth theorem in muh the same way that the onvergene of ψ(α, s) does see [5, Theorem ] An elementary omputation yields ξ(α, s) = 2i (H(α, s,, 0) + 2 ζ(s) ), so rationality statements for H(α, s) for even integral s translate to rationality statements for ξ(α, s) at odd integral s In ontrast, ψ(α, s) does not seem to have a simple expression in terms of H(α, s, p, q) Cruially, however, we still have a relation between ertain speial values of H and ψ relying on the relation 3 = + : ψ ( α 2, s) = = n s e( nα) + e( nα) n s e( nα) e(nα) n s = η(α, s,, 0) η(α, s,, 0) If s = 2k, then e( s ) = and we onlude that 2 (3) ψ( α, 2k) = H(α, 2k,, 0) 2 e( 3nα) e(nα)

4 PIERRE CHAROLLOIS AND MATTHEW GREENBERG For the rest of the paper, suppose that α is a real quadrati irrationality Formulas of Berndt and Arakawa allow us to evaluate H(α, 2k,, 0) rather expliitly Let a b V = SL d 2 (Z) be a matrix suh that Set > 0 and β := α + d > 0 (p, q ) = (p, q)v and ϱ = {q } {p }d Theorem 33 ([, Theorem and Eq (9)]) Suppose that p and p are not in Z Then the following transformation formula holds: β s H(V α, s, p, q) = H(α, s, p, q ) + (2π) s e( s )L(α, s, p, q,, d), where L(α, s, p, q,, d) is as in (2) of [3] If s = m is a negative integer, then L(α, m, p, q,, d) = 2πi m+2 ({ }) m + 2 j {p } jd + ϱ b l b m+2 l ( β) l (m + 2)! l j= l=0 Here, b l is the l-th Bernoulli polynomial By [, Lemma ], there is a totally positive unit β of Q(α) and a matrix V SL 2 (Z) suh that ( ( α α > 0, (p, q ) := (p, q)v (p, q) (mod Z 2 ), and V = β The last ondition implies that β = α + d, onsistently with the notation introdued above Suppose p / Z Then p / Z, too, as p p (mod Z) Applying Theorem 33, observing that H only depends on the lass of (p, q) modulo Z 2, and rearranging terms, we get (β s H(α, s, p, q) = (2π) s e( s )L(α, s, p, q,, d) By the seond part of Theorem 33, if s = 2k then () H(α, 2k, p, q) π 2k = β 2k (2k +! j= m+2 l=0 ( 2k + Setting (p, q) = (, 0) and using (3), we onlude: l ) ({ }) j {p } jd + ϱ b l b 2k+ l ( β) l Theorem 3 Suppose α is a real quadrati irrationality and k is a positive integer Then ψ(α, 2k) Q(α) Moreover, if x x is the nontrivial automorphism of Q(α), then ψ(α, 2k) = ψ(α, 2k) π 2k π 2k π 2k Conjeture follows from Theorem 3 and the evenness of the seant funtion

5 RATIONALITY OF SECANT ZETA VALUES 5 Referenes [] Tsuneo Arakawa, Generalized eta-funtions and ertain ray lass invariants of real quadrati fields, Math Ann 260 (982), 75-9 [2] Tsuneo Arakawa, Dirihlet series ot(πnα)/ns, Dedekind sums, and Heke L-funtions for real quadrati fields, Comment Math Univ St Paul 37 (988), [3] Brue C Berndt, Generalized Dedekind eta-funtions and generalized Dedekind sums, Trans AMS 78 (973), [] Brue C Berndt, Dedekind sums and a paper of G H Hardy, J London Math So (2) 3 (976), [5] Matilde Lalìn, Franis Rodrigue and Matthew Rogers, Seant zeta funtions, J Math Anal Appl 09 (20, no, [6] Mathias Lerh, Sur une série analogue aux fontions modulaires, C R Aad Si Paris, 38 (90), [7] Joseph Lewittes, Analyti ontinuation of Eisenstein series, Trans Amer Math So 77 (972), [8] Bruno Shoeneberg, Ellipti modular funtions (Die Grundlehren der Math Wiss 203), Springer-Verlag (97) address: harollois@mathjussieufr Institut de mathématiques de Jussieu, Université Paris 6, Plae Jussieu, Paris, Frane address: mgreenbe@ualgarya Department of Mathematis and Statistis, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N N, Canada