SOME FORMULAS OF RAMANUJAN INVOLVING BESSEL FUNCTIONS. Henri Cohen

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1 SOME FORMULAS OF RAMANUJAN INVOLVING BESSEL FUNCTIONS by Henri Cohen Abstract. We generalize a number of summation formulas involving the K-Bessel function, due to Ramanujan, Koshliakov, and others. Résumé. Nous généralisons des formules sommatoires faisant intervenir les fonctions K de Bessel, et qui sont dues à Ramanujan, Koshliakov et d autres. 1. Introduction and Tools In his notebooks, Ramanujan has given a number of interesting summation formulas involving the K-Bessel function. These were generalized by Koshliakov and Berndt Lee Sohn in [1]. The purpose of the present paper is to give a general framework for these formulas, and to generalize them. We only give a bare sketch of the proofs. First, we recall that the K-Bessel function K ν with ν C and R >0 can be defined by the formula K ν = 1 t ν 1 e /t+1/t dt. 0 Since K ν = K ν, we will always assume implicitly that Reν 0. These functions possess many important properties, but the most important for us will be their behavior as, and to a lesser etent as 0: as we have π 1/ K ν e note that the right hand side is independent of ν, and as 0 we have K ν ν 1 Γν ν when Reν > 0, and K 0 log. Second, we recall that the Riemann zeta function ζs which is usually defined by the series n s for Res > 1, can be etended to the whole comple plane into a meromorphic function having a single pole, simple, at s = 1, and satisfies the functional equation Λ1 s = Λs, with Λs = π s/ Γs/ζs. Third, we recall that σ s n denotes the sum of the sth powers of the positive divisors of n: σ s n = d n ds.

2 60 Some formulas of Ramanujan involving Bessel functions Fourth, we recall that the real dilogarithm function Li is defined by Li = 0 log 1 t /t dt, so that Li = n /n when 1. This function has many functional equations, and in particular Li + Li 1/ = Cπ /6 log /, where C = for > 0 and C = 1 for < 0, which immediately implies the behavior of Li when. Fifth, we define the standard nonholomorphic Eisenstein Series Gτ, s for Res > 1 and Imτ > 0 by Gτ,s = 1 ys mτ + n s, m,n Z where means that the term m,n = 0,0 must be omitted, and τ = +iy. This function is invariant in τ under the usual action of SL Z, in other words it is a nonholomorphic modular function of weight 0. It is easy to compute its Fourier epansion, which is given by Gτ,s = ζsy s + π1/ Γs 1/ ζs 1y 1 s Γs 1/ πs σ s 1 n + 4y Γs n s 1/ K s 1/πnycosπn. In particular this epansion shows that Gτ,s has a meromorphic continuation to the whole comple s-plane, with a single pole, at s = 1, which is simple with residue π/ note that this is independent of τ, and if we set Gτ,s = π s ΓsGτ,s we have the functional equation Gτ, 1 s = Gτ, s.. The First Theorem and Specializations We implicitly consider that variables such as, b, and so forth are real and strictly positive, in particular different from 0. Theorem.1. We have Λs 1 s/ s 1/ + 4 1/ σ s n n s/ K s/πn =Λ s 1+s/ 1+s/ + 4 1/ σ s n πn n s/ K s/. This immediately follows from the Fourier epansion of Gτ,s. Corollary.. 4 σ s n n s/ K s/ πn = 4 σ s n n s/ K s/πn + Λs 1 s/ s/ + Λs s/ s/.

3 Henri Cohen 61 We now specialize to the first few integer values of s. Corollary.3. Let as usual dn = σ 0 n be the number of divisors of n. We have γ log4π + 4 πn 4π dnk 0 = γ log πn. dnk Corollary.4. We have π 1 + σn n e πn/ = π 1 + σn n e πn log. Both corollaries follow quite easily from the theorem with s = 0 and s = 1 respectively. This second corollary is in fact the transformation formula of the logarithm of Dedekind s eta function. More precisely: Corollary.5. Set L = π 1 + log 1 e πn, which is equal to logηi. We have 1 L Corollary.6. π = L + log σ n n K 1πn = ζ3 1 π + 4 σ n n K 1. πn. This is the theorem for s =. For s 3 odd, we obtain from the theorem formulas which are closely linked to the theory of Eichler Shimura of k -fold integrals of modular forms of weight k, giving pseudo-modular forms of weight k. For instance, for s = 3 we obtain directly: Corollary σ 3 n n ζ3 π = π e πn πn σ 3 n n 1 + e πn/. πn However, as mentioned above, this is not the point. A consequence is the following: Corollary.8. Set H = π π3 7 + ζ3 + σ 3 n n 3 e πn.

4 6 Some formulas of Ramanujan involving Bessel functions Then H = π 3 /30E 4 i, where E 4 τ is the usual holomorphic Eisenstein series of weight 4, and H satisfies the functional equation 1 H = 1 H, in other words is a pseudo-modular form of weight. This is quite easily proved by integration, ecept that there is an a priori unknown constant of integration, which in the above corollary is the coefficient of. To compute this constant eplicitly, we need to compute the asymptotic epansion of both sides, and after some work we obtain the above formula. As mentioned above, this type of formula eists for all odd integral values of s, with the series σ sn/n s e πn in the right-hand side. 3. The Second Theorem and Specializations 3.1. Integral Representations. Note the following integral representations of the sums that we will consider: Proposition 3.1. For > 0 and all s C we have dt 0 t s+1 e πt 1e π/t 1 = s/ n s/ σ s nk s 4π n. Corollary 3.. Under the same conditions, for all j 0 we have P j e πt 0 t s+1 e πt 1 j+1 e π/t 1 dt = 1j s/ n s/+j σ s j nk s 4π n, where P j X is the sequence of polynomials defined by induction by P 0 X = 1 and P j X = XX 1P j 1 X jxp jx for j 1. Corollary 3.3. We have log 1 e πt 0 t s+1 e π/t 1 = s/ n s 3/ σ 1 s nk s 4π n. These results are proved by replacing K s by its integral definition, and then integrating several times. The main results of this section give formulas for the right hand side of the proposition. 3.. The Case s / Z. Theorem 3.4. Let s / Z such that Res 0. For any integer k such that k Res + 1/ we have the identity 8π s/ n s/ σ s nk s 4π n = As,ζs + Bs,ζs ζiζi s i 1 + k+1 σ s n ns k s k sinπs/ n, 1 i k

5 Henri Cohen 63 where As, = s 1 sinπs/ π1 s Γs and Bs, = π s 1 Γs + 1 π s cosπs/. Remarks. 1 In the above theorem, if = n Z 1 the epression n s k s k /n is of course to be interpreted as its limit as tends to n, in other words as s/ kn s k. The condition Res 0 is not restrictive since the left hand side of the identity is invariant under the change of s into s. 3 Although the theorem is valid when k Res+1/, the convergence of the series on the right hand side is as fast as possible without acceleration only when k Res/, in other words k Res + / if Res / Z. The proof of the theorem involves integrating the formula of the theorem of the preceding section, and doing a careful etension process to show that it is valid for all s as given. The specializations of the above theorem to s = 1/ and s = 3/ are as follows: Corollary 3.5. π σ 1/ ne 4π n = σ 1/ n n 1/ + 1/ n π π1/ ζ3/ π 1 1/ ζ1/. Corollary 3.6. σ 3/ n 3/ n4π σ n + 1e 4π n = 4 3 n 1/ + n 1/ n π + π3/ ζ5/ + 1/ 1 ζ3/ + π 3 ζ1/ The Case s Z. As usual, we assume that s 0. The result corresponding to s = m with m Z 1 is as follows: Theorem 3.7. Let m Z 1. We have the identity 8 1 m σ m n n m K m4π n = 4 m+1 π σ m n logn/ n m n 1 i m 1 + π m 1 ζmlogπ + ζ m + B m m m + 1 m 4m! π m+ m ζm + 1. m+1 ζiζ i m i m 1

6 64 Some formulas of Ramanujan involving Bessel functions Note that by the functional equation, if desired we can replace ζ i m by its epression in terms of ζm + 1 i. For instance, by simple replacement, for s =, in other words for m = 1, we obtain Corollary 3.8. We have the identity 8 σ n n K 4π n = 4 π σ n logn/ n n logπ + ζ ζ The result corresponding to s = m = 0 is as follows: π 4 + ζ3. Theorem 3.9. We have 4 dnk 0 4π n = π dn logn/ n γ + log + logπ1/ π Here again we need to do a careful limiting process. Corollary π 3 1/ n 1/ dnk 1 4π n = dnlogn/ n + n dn 1 n + 1 logπ 1/ 1 4 π. Simply compute derivatives The Case s 1 + Z. Once again we assume that s 0. The result corresponding to s = m + 1 with m Z 1 is as follows: Theorem Let m Z 1. We have the identity 8π 1 m 1/ σ m+1 n n m+1/ K m+14π n = m+ σ m+1 n n m+ n + + ζiζi m 1 i m 1 1 i m + m + 1 m+1 m! π m m ζm m+1 log + γζm + + ζ m + The result corresponding to m = 0, in other words to s = 1, is as follows: B m+ m + 1 m+1.

7 Henri Cohen 65 Theorem 3.1. We have the identity 8π 1/ σn n 1/ K 14π n = σn n n + + logπ + γ log + γζ + ζ, where as usual σn = σ 1 n. Corollary We have the identities 8π σnk 0 4π n = 1 n σn 1 n + and 8π 3 1/ 16π 4 π log + γζ + ζ, n 1/ σnk 1 4π n = nσnk 0 4π n = σn n π , σn n n Asymptotics of Sums σ mnf/n. Set T m f = σ mnf/n. A careful study of these sums for a large and useful class of functions f is necessary, so as to obtain their asymptotic epansions as. This is quite nontrivial, and requires several pages of computations. We only give the results for the functions that we need: Eample 1. ft = log1 + t t and m. In that case: T m f = ζm + 1log + ζ m γζm + 1 ζm/log ζ m/ logπ/ζm + o1. Eample. ft = logtlog 1 t and m. In that case: T m f = π /ζm + 1 ζmlog ζ m + logπζmlog ζ m logπζ m + ζ 0ζm + o1. Eample 3. ft = Li t and m. In that case: T m f = ζmlog + ζ m + logπζmlog + ζ m + logπζ m ζ 0 + π /6ζm + o1. 4. Integration of the Formulas for s Z We can integrate the results that we have obtained for m Z. Integration term by term is trivially justified, but the whole difficulty lies in the determination of the constant of integration. This is done by comparing the asymptotic epansions of both sides, that of the right-hand side being obtained thanks to the study mentioned above, and in particular of the specific eamples.

8 66 Some formulas of Ramanujan involving Bessel functions 4.1. The Case s Z, s 0. Theorem 4.1. Let m Z 1. We have the identity 4 π m+1/ 1 m 1 1 π i m 1 σ m n n m+1/k m 14π n = σ m n n m log log n ζiζ i m i m m i 1 + Li + π ζmlogπ + ζ mlog + ζm log B m mm 1 m 1 + m 1 4m 1! 1 π m+ m ζm C m, where C m is a constant given by C m = ζ m + logπζ m + π /6 ζ 0ζm. n n As mentioned above, the only difficulty is in the computation of C m, which is done by making and comparing asymptotic epansions. 4.. The Case s = 0. Theorem 4.. dn π 1/ n 1/ K 14π n = 1 π dn log log 1 n n + Li n γ 1 + log + logπ π log + log 4π + C 0 where C 0 is an eplicit constant which I have not had the time to compute The Case s = m + 1, m 1.

9 Henri Cohen 67 Theorem m+1 m = + + σ m+1 n n m+1 K m 4π n σ m+1 n n m+1 log 1 + n n ζi m 1 ζi i m 1 i m 1 1 i m m m 1! log + 1 π m m ζm + 1 B m+ m + 1m + m+1 + logζm + + γ 1ζm + + ζ m + + C m+1 = + + logγ1 + /n n m+1 ζi m 1 ζi i m 1 i m 1 1 i m m m 1! log + 1 π m m ζm + 1 B m+ m + 1m + m+1 + logζm + + ζm + + ζ m + + C m+1 where C m+1 is a constant given by 4.4. The Case s = 1. Theorem 4.4. We have the identity where C 1 is a constant given by C m+1 = ζ m logπζm + 1. σn n K 04π n = σn n log + logπ + γ log 1 + n n log + C π 6 log + γ 1ζ + ζ C 1 = 5π 48 + log π + γ logπ γ 4 4 γ 1 = π 16 log π + γ logπ + γ 4 4 ζ 0, where γ 1 = lim N 1 n N logn/n log N/.

10 68 Some formulas of Ramanujan involving Bessel functions For the convenience of the reader, here are some numerical values: γ 1 = ζ 0 = C 1 = References [1] B. Berndt, Y. Lee, and J. Sohn, Koshliakov s formula and Guinand s formula in Ramanujan s lost notebook, in K. Alladi ed., Surveys in number theory, Springer 008, p [] H. Cohen, Number Theory vol I: Tools and Diophantine Equations, Graduate Tets in Math. 39, Springer-Verlag 007. [3] H. Cohen, Number Theory vol II: Analytic and Modern Tools, Graduate Tets in Math. 40, Springer-Verlag 007. January 17, 010 Henri Cohen, Université Bordeau I, Institut de Mathématiques, U.M.R. 551 du C.N.R.S, 351 Cours de la Libération, TALENCE Cede, FRANCE henri.cohen@math.u-bordeau1.fr