1. Introduction 1.1. Background and previous results. Ramanujan introduced his zeta function in 1916 [11]. Following Ramanujan, let.

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1 IDENTITIES FOR THE RAMANUJAN ZETA FUNCTION MATHEW ROGERS Abstract. We prove formlas for special vales of the Ramanjan ta zeta fnction. Or formlas show that L(, k) is a period in the sense of Kontsevich and Zagier when k 2. As an illstration, we redce L(, k) to explicit integrals of hypergeometric and algebraic fnctions when k {2, 3, 4, 5}.. Introdction.. Backgrond and previos reslts. Ramanjan introdced his zeta fnction in 96 []. Following Ramanjan, let (z) := q ( q n ) 24 = n= τ(n)q n, where q = e 2πiz. Ramanjan observed that τ(n) is mltiplicative, and this lead him to stdy the Dirichlet series: L(, s) := n= n= τ(n) n s. Mordell sbseqently proved that L(, s) has an Eler prodct, and satisfies the fnctional eqation (2π) s 2 Γ(2 s)l(, 2 s) = (2π) s Γ(s)L(, s) [9], [2, pg. 242]. The main goal of this paper is to prove formlas for L(, k) when k is a positive integer. Kontsevich and Zagier defined a period to be a nmber which can be expressed as a mltiple integral of algebraic fnctions, over a domain described by algebraic eqations [8]. The ring of periods contains both the algebraic nmbers, and certain transcendental nmbers like π and log 2. It follows from the work of Beilinson [2], and Deninger and Scholl [7], that special vales of L-fnctions attached to modlar forms are also periods. Paraphrasing [8, p. 24], their reslts follow from deep cohomological maniplations, and a carefl stdy of vales of reglators. Following Deligne, we say that L(, k) is a critical L-vale if k. Kontsevich and Zagier smmarized the properties of critical L-vales in [8]. It is relatively easy to relate these vales to integrals of algebraic fnctions. The standard Mellin Date: March 28, Mathematics Sbject Classification. Primary 33C2; Secondary F, F3, Y6, 33C75, 33E5. Key words and phrases. Ramanjan zeta fnction, τ Dirichlet series, L-vales.

2 2 MATHEW ROGERS transform gives L(, k) = (2π)k (k )! k (i)d () whenever k. The sal method for obtaining an elementary integral, is to set = n= F ( α) 2F (α), (2) where F (α) is the classical hypergeometric series: ( ) 2 2n ( α ) n F (α) := (3) n 6 = 2 π d ( 2 ) ( α 2 ). (4) Notice that 2i is the period ratio of the elliptic crve y 2 = ( x 2 )( αx 2 ). Now appeal to the classical formlas [3, pg. 24], [3, pg. 2]: (i) = 6 α( α)4 [F (α)] 2, (5) d dα = 2πα( α) [F (α)] 2, (6) and notice that α (, ) when (, ). Eqation () redces to L(, k) = π k 6(k )! ( α) 3 [F (α)] k [F ( α)] k dα. (7) Sbstitting (4) allows s to dedce that π k L(, k) is a period if k. Kontsevich and Zagier illstrated this point with an eqivalent identity [8, pg. 24]..2. Main reslts. It is not at all obvios that L(, k) is a period if k 2. The method from the previos section fails, becase the integrand in eqation (7) becomes a ratio of algebraic and hypergeometric fnctions. It seems to be very difficlt to pass from eqation (7) to anything interesting. In this paper we will se ideas from the philosophy established jointly with Zdilin [6], [7], [22], to redce these L-vales to integrals of algebraic and hypergeometric fnctions. For example, we prove the following theorem: Theorem. The following identity is tre: L(, 2) = 28π 824! where F (α) is defined in (3). [F (α)f ( α)] 5 ( ) α + 876α α 3 + 2α 4 α log α dα, (8)

3 IDENTITIES FOR THE RAMANUJAN ZETA FUNCTION 3 Formla (8) shows that L(, 2) belongs to the ring of periods. Most of or identities involve hypergeometric fnctions, bt these always redce to elliptic integrals by (4). Eqation (8) becomes a massive twelve dimensional integral: L(, 2) = 52π [ d dz ( 2 )( z 2 )( α 2 )( ( α)z 2 ) ( ) α + 876α α 3 + 2α 4 α α t dt dα. (9) ] 5 Theorem 2 provides the key formlas we need to express L(, k) in terms of two dimensional integrals of algebraic and hypergeometric fnctions for k 2. Corollary highlights examples when k {3, 4, 5}. The k = 2 case is apparently the only instance where a redction to a one-dimensional integral is possible, and we discss this case separately in Section 2. Finally, we note that eqation (8) is closely related to the moments of elliptic integrals stdied in [], [9], and [2]. 2. A formla for L(, 2) The main goal of this section is to prove a formla for L(, 2) sing the method developed in [6] and [7]. In Section 3 we stdy L(, k) for arbitrary k 2. The crcial first step is to decompose (z) into a linear combination of prodcts of two Eisenstein series. The sal Eisenstein series is defined by E k (z) := + 2 ζ( k) n= n k e 2πinz e 2πinz. The most famos decomposition of (z) is de to Ramanjan: 728 (z) = E 4 (z) 3 E 6 (z) 2, () bt this formla involves E 4 (z) 3, and the method from [6] and [7] only applies to modlar forms which decompose into prodcts of two Eisenstein series. We avoid this obstrction by considering a linear combination of s. Lemma. We have (z) + 24 (2z) + 2 (4z) = [E 6(2z) 64E 6 (4z)] [E 6 (z) 33E 6 (2z) + 32E 6 (4z)]. Proof. It is easy to show that both sides of the eqation are modlar forms on Γ (4). By the standard valence formla for congrence sbgrops [8], the two sides are eqal if their Forier series expansions agree to more than [Γ() : Γ (4)] = 6 terms. We sed a compter to check that the first Forier coefficients agree, and ths we conclde that the identity is tre. It is possible to constrct an alternative elementary proof by applying formlas from Ramanjan s notebooks (se [3, pg. 24, Entry 2] and [3, pg. 26, Entry 3]). ()

4 4 MATHEW ROGERS Now apply an involtion to the first term on the right-hand side of (). The eqation becomes [ ( ) ( )] (z) + 24 (2z) (4z) = E 54 2 (2z) 6 6 E 6 2z 4z. [E 6 (z) 33E 6 (2z) + 32E 6 (4z)]. Sppose that z = i with. The right-hand side redces to a for-dimensional infinite series. We have (i) + 24 (2i) + 2 (4i) = 8 6 n,m,r,s m,r,s odd (nr) 5 nm e 2π( +rs). Mltiply both sides by k and integrate for (, ). By niform convergence: 8 ( k + 2 2k) L(, k) = (2π)k (k )! n,m,r,s m,r,s odd (nr) 5 k 7 nm e 2π( +rs) d. The rational term on the left cancels the Eler factor of L(, s) at the prime p = 2. Notice that ( k + 2 2k) τ(n) L(, k) = n. s n= n odd Finally apply the key trick: Use a change of variables to swap the indices of smmation inside the integral in (2). If the trick is properly exected, then the right-hand side redces to an integral involving modlar fnctions. Proof of Theorem. Set k = 2 and let n/r in (2). The formla becomes 824 L(, 2) = 2 22 π2! =! 5 r,m r,m odd 5 log 2πrm r e m= m odd n e 2πns d n,s s odd + e 2πm e 2πm Now sppose that and α are related by (2). In particlar: n= n e 2πn d. e 4πn (2) = F ( α) 2F (α).

5 IDENTITIES FOR THE RAMANUJAN ZETA FUNCTION 5 Then α (, ) when (, ). We compte d/dα sing (6), and varios identities from Ramanjan s notebooks imply + e 2πm = α /8, (3) e 2πm m= m odd n= n e 2πn e 4πn = 32 α ( α + 876α α 3 + 2α 4) [F (α)] 2. (4) We can prove (3) sing [3, pg. 24, Entry 2], and the proof of (4) follows from the Eisenstein series relation E 2 (z) = E 6 (z) 2 and [3, pg. 26, Entry 3]. Combining the varios formlas completes the proof. 3. Formlas for L(, k) when k 2 In this section we obtain doble integrals for L(, k) when k 2. In general, it seems to be difficlt to simplify formlas for L(f, k), when k > weight(f). So far there is only one instance where sch an L-vale has been redced to recognizable special fnctions. Zdilin proved that ( ) ( π L(g, 3),,, 3/2 =Γ2 2 (/4) 4 F 3 7 ; + 24Γ 2,,, (3/4) 4, 3 2, F , 3 2, 3 2 ( ) + 3Γ 2,,, 2 (/4) 4 F 3 ;. 3 4, 3 2, 3 2 where g(z) = η 2 (4z)η 2 (8z) is the weight 2 CM newform attached to condctor 32 elliptic crves [22], [23]. Rodrigez-Villegas and Boyd sed nmerical experiments to find many relations between Mahler measres and vales of L(f, k), bt all of their conjectres are still open [5], [4]. Theorem 2. Assme that k 2. If k is odd: 8 ( k + 2 2k) L(, k) where If k is even: = 2k 7 i k π 2k [ζ(6 k)] 2 (k )!(k 2)! 8 ( k + 2 2k) L(, k) ) ; (z ) k 2 P k 5 ()Q k 5 (z)dz d, (5) (6) P k () :=E k (i) (2 + 2 k )E k (2i) + 2 k+ E k (4i), (7) Q k (z) :=E k (iz) E k (2iz). (8) = 2k i k π 2k ζ( k)ζ( k) (k )!(k 2)! (z ) k 2 5 R k ()S k (z)dz d, (9)

6 6 MATHEW ROGERS where R k () :=E k (2i) 2 k E k (4i), (2) S k (z) :=E k (iz) ( + 2 k )E k (2iz) + 2 k E k (4iz). (2) Proof. We prove (6) first. Assme that k is odd in (2), and let m/r. The integral becomes Let, then 8 ( k + 2 2k) L(, k) = (2π)k (k )! k 7 m m odd m k 6 e 2πm e 4πm n,r r odd n 5 2πnr e rk d. 8 ( k + 2 2k) L(, k) = (2π)k (k )! 4() 5 k m m odd m k 6 e 2πm e 4πm n,r r odd n 5 r k e 2πnr d. (22) Using the involtion for E k (), we easily find that 4() 5 k m m odd m k 6 e 2πm e 4πm = i k 2 5 k ζ(6 k)p k 5 (), (23) where P k () is defined in (7). To simplify the second sm in (22), we reqire the integral: e 2πa a s If s k and a nr, then n,r r odd = (2π)s Γ(s) n 5 r k e 2πnr = n,r r odd = (2π)k (k 2)! (z ) s e 2πaz dz. (24) n k 6 e 2πnr (nr) k = (2π)k ζ(6 k) 2(k 2)! (z ) k 2 n,r r odd n k 6 e 2πnrz dz (z ) k 2 Q k 5 (z)dz, (25) where Q k (z) is defined in (8). Finally combine (25), (23), and (22) to complete the proof of (6).

7 IDENTITIES FOR THE RAMANUJAN ZETA FUNCTION 7 Next we prove (9). The steps are similar to the proof of (6), so we will be brief. Assme that k is even in eqation (2), and let n/r. Then we find 8 ( k + 2 2k) L(, k) = (2π)k (k )! = (2π)k (k )! = (2π) 2k (k )!(k 2)! k 7 n,s s odd 4() 5 k n,s s odd n k e 2πns n k e 2πns m,r m,r odd m,r m,r odd e rk 2πmr d r k e 2πmr d, (z ) k 2 4() 5 k n,s s odd m,r m,r odd n k e 2πns m k e 2πmrz dz d. The second eqality follows from mapping, and the third eqality follows from (24). Finally, it is easy to show that 4() 5 k n,s n k e 2πns = i k 2 k ζ( k) 5 R k (), m,r m,r odd s odd m k e 2πmrz = 2 ζ( k)s k (z), where R k () and S k (z) are defined in (2) and (2).

8 8 MATHEW ROGERS Corollary. Let F (α) denote the sal hypergeometric fnction, defined in (3). The following identities are tre: L(, 3) = π3 q 3 L(, 4) = π5 q 4 L(, 5) = π7 q 5 α [F (α)f ( β) F (β)f ( α)] [F (α)f (β)] 5 ( + α)(7 32α + 7α2 )(2 + 3β + 2β 2 ) dβ dα, (26) α( β) α [F (α)f ( β) F (β)f ( α)] 2 [F (α)f ( α)] 5 (2 α) ( α α2 52α 3 + α 4 ) (2 β) dβ dα, α( β) (27) α [F (α)f ( β) F (β)f ( α)] 3 [F (α)f (β)] 5 (3 47α + 33α2 47α 3 + 3α 4 ) ( + β) ( + 29β + β 2 ) dβ dα, α( β) (28) where q 3 = , q 4 = , and q 5 = /4. Proof. We sketch the proof of eqations (26) and (28) below. Set = and then apply (6). Eqation (6) becomes 8 ( k + 2 2k) L(, k) = 2k 7 i k π 2k [ζ(6 k)] 2 (k )!(k 2)! α ( F ( β) 2F (β) Finally we se the formlas F ( α) F ( β), z = 2F (α) 2F (β), ) k 2 F ( α) P k 5 ()Q k 5 (z) dβ dα 2F (α) 4π 2 α( α)β( β)f (α) 2 F (β). 2 P 8 () =5 ( α 2) ( 7 32α + 7α 2) [F (α)] 8, Q 8 (z) =5β ( 2 + 3β + 2β 2) [F (β)] 8, P () =33( α) ( 3 47α + 33α 2 47α 3 + 3α 4) [F (α)], Q (z) = 33 2 β( + β) ( + 29β + β 2) [F (β)], when k = 3 and k = 5. These formlas are easy to prove by expressing E k (i) in terms of polynomials in E 4 (i) and E 6 (i), and then appealing to [3, pg. 26, Entry 3]. In practice, we sed nmerical searches to find linear dependencies between P k (), Q k (), and {α j [F (α)] k} j=k. j=

9 IDENTITIES FOR THE RAMANUJAN ZETA FUNCTION 9 If and α are related by (2), then it is a classical fact that E k (2 j i) = (polynomial in α) [F (α)] k for j {,, 2} and k even. This makes it possible to see that the pattern of Corollary contines. The identities for L(, k) are always two dimensional integrals containing [F (α)f ( β) F (β)f ( α)] k 2. Frthermore, if k is even the integral contains [F (α)f ( α)] 5, and if k is odd the integral contains [F (α)f (β)] Speclation and Conclsion We have proved that L(, k) is a period for k 2. It is interesting to speclate on what simplifications might be possible for the integrals in Theorem and Corollary. We can draw an analogy with the case of weight two modlar forms. If we select a specific modlar form sch as f(z) = η(z)η(3z)η(5z)η(5z), then it is possible to prove reslts like 5 L(f, 2) = 4π2 log + X + X + Y + Y dt ds, (29) where X = e 2πit and Y = e 2πis. The integral on the right is a Mahler measre, and the srface obtained from setting +X +X +Y +Y =, is precisely the elliptic crve attached to f(z) by the modlarity theorem. Rodrigez-Villegas gave a very nice explanation of why reslts like (29) exist [3], [6], [4], [7]. The key point for s, is that the proof of (29) does not reqire any prior knowledge of the elliptic crve attached to f(z). Ths it seems plasible that there might be an analogos relation between L(, 2)/π 2 and the Mahler measre of a polynomial in 2 variables. If sch an identity exists, then it shold be possible to derive it from eqation (8) with only calcls. Frthermore, the 2 variable polynomial shold give an affine model of a hypersrface attached to (z). Deligne fond sch a hypersrface in his proof of the Ramanjan-Petersson conjectres, bt his hypersrface is typically described sing étale cohomology. Acknowledgements. The athor thanks Wadim Zdilin for his encoragement and sefl sggestions. References [] J. M. Borwein, D. Borwein, M. L. Glasser, and J. Wan, Moments of Ramanjan s generalized elliptic integrals and extensions of Catalan s constant, J. Math. Anal. Appl. 384 (2), no. 2, [2] A. Beilinson, Higher reglators of modlar crves, in: Applications of Algebraic K theory to Algebraic Geometry and Nmber Theory. Contemp. Math. 55, Amer. Math. Soc., Providence (986), 34. [3] B. C. Berndt, Ramanjan s Notebooks, Part III (Springer-Verlag, New York, 99). [4] D. W. Boyd, Mahler s measre and special vales of L-fnctions, Experiment. Math. 7 (998), [5] D. Boyd, Mahler s measre and special vales of L-fnctions of elliptic crves at s = 3, Preprint (26).

10 MATHEW ROGERS [6] C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Z n - actions, J. Amer. Math. Soc. (997), no. 2, [7] C. Deninger and A. Scholl, The Beilinson conjectres, in: L-fnctions and Arithmetic, London Math. Soc. Lectre Notes 53 (eds. J. Coates and M. J. Taylor), Cambridge Univ. Press, Cambridge (99), [8] M. Kontsevich and D. Zagier, Periods, in: Mathematics nlimited 2 and beyond (Springer, Berlin, 2), [9] L. J. Mordell, On Mr. Ramanjan s Empirical Expansions of Modlar Fnctions, Proc. Cambridge Phil. Soc. 9 (97), [] K. Ono, The Web of Modlarity: Arithmetic of the Coefficients of Modlar Forms and q-series, American Mathematical Society, Providence, RI, 24. [] S. Ramanjan, On certain Arithmetical Fnctions, [Trans. Camb. Phil. Soc. 2 (96), no. 9, 59 84]. Collected papers of Srinivasa Ramanjan, 23-29, AMS Chelsea Pbl., Providence, RI, 2. [2] R. A. Rankin Modlar Forms and Fnctions, Cambridge University Press, Cambridge, 977. [3] F. Rodrigez-Villegas, Modlar Mahler measres I, in: Topics in nmber theory (University Park, PA, 997), Math. Appl. 467 (Klwer Acad. Pbl., Dordrecht, 999), [4] F. Rodrigez-Villegas, R. Toledano and J. D. Vaaler, Estimates for Mahlers measre of a linear form Proc. Edinb. Math. Soc. 47 (24), [5] M. D. Rogers, New 5 F 4 hypergeometric transformations, three-variable Mahler measres, and formlas for /π, Ramanjan J. 8 (29), no. 3, [6] M. Rogers and W. Zdilin, From L-series of elliptic crves to Mahler measres, Compositio Math. 48 (22), no. 2, [7] M. Rogers and W. Zdilin, On the Mahler measre of + X + /X + Y + /Y, to appear in Int. Math. Res. Notices. [8] B. Schoeneberg Elliptic modlar fnctions: an introdction. Springer-Verlag, New York, 974. Translated from the German by J. R. Smart and E. A. Schwandt, Die Grndlehren der mathematischen Wissenschaften, Band 23. [9] J. G. Wan, Moments of prodcts of elliptic integrals, Adv. Appl. Math. 48 (22), no., 2 4. [2] D. Zagier Introdction to modlar forms. in: From Nmber Theory to Physics, M. Waldschmidt et al, Springer-Verlag, Heidelberg (992) [2] Y. Zho, Legendre Fnctions, Spherical Rotations, and Mltiple Elliptic Integrals, preprint arxiv: [math.ca] (23). [22] W. Zdilin, Period(d)ness of L-vales, Proceedings of the International Conference in memory of Alf van der Poorten (to appear), J.M. Borwein et al. (eds.), 4 pages. [23] W. Zdilin, Hypergeometric evalations of L-vales of an elliptic crve, A plenary talk, Ramanjan-25 Conference The Legacy of Srinivasa Ramanjan (University of Delhi, New Delhi, India, December 722, 22) Department of Mathematics and Statistics, Université de Montréal, CP 628 scc. Centre-ville, Montréal Qébec H3C 3J7, Canada address: mathewrogers@gmail.com