GENERALIZED LAMBERT SERIES AND ARITHMETIC NATURE OF ODD ZETA VALUES ATUL DIXIT AND BIBEKANANDA MAJI 1 X nN−2h Abstract. It is pointed out that the generalized Lambert series studied by enN x − 1 n=1 Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Note- book in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parame- ters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m 2 Z. It generalizes a transformation of 1 Wigert and can be regarded as a formula for ζ 2m + 1 − N . Applications of these trans- formations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ. 1. Introduction Surprises from Ramanujan's Lost Notebook [31] seem to continue unabated. We recently came across a new instance of this phenomenon while reading page 332 of the Lost Notebook, which we describe soon. Page 332 is mainly devoted to two versions of a beautiful formula for 1 ζ 2 , one of which is also given in Entry 8 of Chapter 15 of Ramanujan's second notebook [29], [6, p. 314], and is as follows. Let α and β be two positive numbers such that αβ = 4π3. Then 1 p ( 1 p p p ) X 1 π2 1 β 1 X cos( nβ) − sin( nβ) − e− nβ = + + ζ + p p p : (1.1) en2α − 1 6α 4 4π 2 n cosh( nβ) − cos( nβ) n=1 n=1 As mentioned in [8, p. 191], one can conceive this result as an identity for an infinite sum of theta functions. Katsurada [23] interpreted the infinite series on the right as the exact form of the error term in the asymptotic formula 1 p X 1 π2 1 β 1 = + + ζ + o(1) en2α − 1 6α 4 4π 2 n=1 2010 Mathematics Subject Classification. Primary 11M06; Secondary 11J81. Keywords and phrases. Lambert series, Dedekind eta function, odd zeta values, Ramanujan's formula, transcendence. 1 2 ATUL DIXIT AND BIBEKANANDA MAJI as α ! 0+. The result (1.1) has instigated a flurry of activities over the past two decades with several mathematicians obtaining generalizations, including analogues for the Hurwitz zeta function and Dirichlet L-functions. A complete account of these activities is given in [8, p. 191-193] and while we avoid duplicating all references given there, we do mention those which are relevant to the material in the sequel. Ramanujan's second reformulation of (1.1) in [31, p. 332] appears in a footnote on page 9 of a paper of Wigert [34] who also generalized it [34, p. 8-9, Equation (5)] by deriving a 1 formula for ζ N for any even positive integer N without knowledge that the case N = 2 1 had been considered by Ramanujan. This alternative reformulation of the formula for ζ 2 has been recently generalized in a different direction in [9, p. 859, Theorem 10.1]. Wigert's formula for N even and x positive1 reads 1 X 1 ζ(N) − 1 1 1 1 = + x N Γ 1 + ζ + enN x − 1 x N N 4 n=1 N 1 1 N 2 −1 −1 N N (−1) 2 2π X iπ(2j+1)(N−1) 2π − (2j+1)πi + e 2N L 2π e 2N N x N x j=0 1 N − iπ(2j+1)(N−1) 2π (2j+1)πi + e 2N L 2π e 2N ; (1.2) N x 1 −1 P1 n N where LN (x) := n=1 1 : exp(n N x)−1 Kanemitsu, Tanigawa and Yoshimoto systematically studied various analogues and gener- alizations of (1.1) which give values of the Riemann zeta function [19, Section 3], Hurwitz zeta function [21, Theorem 2.1], and also Dirichlet L-functions [22, p. 32-33] at rational arguments with even denominator and odd numerator. The culmination of their results occurred in [20, Theorem 1] when they obtained the following beautiful result for values of the Riemann zeta function at positive rational arguments irrespective of the parity of the numerator and the denominator. Let h and N be fixed natural numbers with h ≤ N=2. Let πj πj a = cos ; b = sin ;A (y) = π (2πy)1=N : (1.3) j;N 2N j;N 2N N Also, let −A n e N ( x ) f0(x; n; N) = n (1.4) 2 sinh AN x and for j ≥ 1, π(2h−1)j n π(2h−1)j n −2AN ( x )aj;N cos 2AN x bj;N + 2N − e cos 2N fj(x; n; N; h) = n n : (1.5) cosh 2AN x aj;N − cos 2AN x bj;N 1Wigert actually obtains the transformation for complex x with Re(x) > 0, however, this can be easily seen to be true by analytic continuation. GENERALIZED LAMBERT SERIES AND ARITHMETIC NATURE OF ODD ZETA VALUES 3 Then for x > 0, 1 X nN−2h = P (x) + S(x); (1.6) enN x − 1 n=1 where 1 ζ(2h) P (x) = P (x; N; h) = − ζ(−N + 2h) + 2 x 1 N − 2h + 1 N − 2h + 1 − (N−2h+1) + Γ ζ x N ; (1.7) N N N and2 N−1 N−2h+1 1 2 (−1)h+1 2π N X 1 X S(x) = S(x; N; h) = f (x; n; N) + f (x; n; N; h) N x 2h−1 0 2j n=1 n N j=1 (1.8) for N odd, and N N−2h+1 1 2 (−1)h+1 2π N X 1 X S(x) = S(x; N; h) = f (x; n; N; h) (1.9) N x 2h−1 2j−1 n=1 n N j=1 for N even. Note that Ramanujan's formula (1.1) is the special case h = 1;N = 2 of the above result. Now the surprising thing that we came across while going over page 332 of the Lost Notebook [31] is that at the end of this page Ramanujan starts writing the exact same series considered by Kanemitsu, Tanigawa and Yoshimoto in their above result, that is, the series on the left side of (1.6), but with more general conditions on the associated parameters subsuming the ones given by Kanemitsu et al. The precise sentence at the end of this page, in Ramanujan's own words, reads 1r 2r 3r + + + ··· (1.10) e1sx − 1 e2sx − 1 e3sx − 1 where s is a positive integer and r − s is any even integer. Note that Ramanujan is taking r and s such that r − s is any even integer, where as for the series on the left side of (1.6), Kanemitsu et al., in Ramanujan's notation, take r − s to be a negative even integer only. Ramanujan does not give any expression for this series like the one in (1.6), so it is not clear what he had in his mind. Was there a page after page 332 in the Lost Notebook that went missing? While this question may remain unanswered forever and while some may call our speculation as wishful thinking, from the fact that the left-hand side of (1.1) is merely a special case of (1.10), it is clear that Ramanujan had recognized the importance of the latter. Perhaps if he had completed his formula, he would have ended up with an expression consisting of the Riemann zeta function at rational arguments. 2There is a minor typo in the statement of this theorem in [20] in that the variable of summation in both the series representations for S(x) there begins from n = 0. See (1.8) and (1.9) here for the corrected expressions. 4 ATUL DIXIT AND BIBEKANANDA MAJI Even though Ramanujan did not give any expression for the series in (1.10), he is right in making the assumption that r−s is any even integer, for, (1.6) holds not only for 0 < h ≤ N=2 but also when h is any integer. This, in Ramanujan's notation, is equivalent to saying r − s can be any even integer. This observation has two fruitful consequences that went unnoticed in the paper [20] of Kanemitsu et al. Indeed, if we allow h to be any negative integer, then the case N = 1; h = 1 − m, where m > 1 is a natural number, of (1.6) gives an interesting result of Ramanujan [29, vol. 1, p. 259, no. 14], [28, p. 269], [30, p. 190], namely, for α; β > 0 with αβ = π2, 1 2m−1 1 2m−1 X n X n B2m αm − (−β)m = (αm − (−β)m) ; (1.11) e2αn − 1 e2βn − 1 4m n=1 n=1 th where Bm is the m Bernoulli number defined by 1 n x X Bnx = ; (jxj < 2π); ex − 1 n! n=1 whereas the case N = 1; h = 0 of (1.6) gives another result of Ramanujan [29, Ch. 14, Sec. 8, Cor. (i)], [31, p. 318, formula (23)], namely, for α; β > 0 such that αβ = π2, 1 1 X n X n α + β 1 α + β = − : (1.12) e2nα − 1 e2nβ − 1 24 4 n=1 n=1 However, (1.11) and (1.12) are but special cases of Ramanujan's following famous formula for ζ(2m + 1) [29, p.