Circular summation of theta functions in Ramanujan's Lost Notebook

On Ramanujan's formula for ζ(1/2) and ζ(2m + 1)

In this note, we shall give a partition-theoretic interpretation which explains the non-negativity of the coefficients of some q-series arising from Ramanujan's Lost Notebook and prove the almost-increasingness of the coefficients via a vector partition generating function.

Let f(a, b) denote Ramanujan's symmetric theta function. In his Lost Notebook, Ramanujan claimed that the “circular” summation of n-th powers of f satisfies a factorization of the form f(a, b)F(ab). He listed elegant identities for n = 2, 3, 4, 5 and 7. We present alternative proofs of his claims.

Extended higher Herglotz function II

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

One page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

Many remarkable cubic theorems involving theta functions can be found in Ramanujan's Lost Notebook. Using addition formulas, the Jacobi triple product identity and the quintuple product identity, we establish several theorems to prove Ramanujan's cubic identities.

Koshliakov zeta functions I: Modular relations

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook 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 parameters 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 ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations 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.

In his lost notebook, Ramanujan recorded several modular equations of degree 5 related to the Rogers-Ramanujan continued fraction R(q). We prove several of these identities and give factorizations of some of them in this paper.

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.

On page 196 in his lost notebook, S. Ramanujan offers evaluations of two particular Dirichlet series. In this paper, we establish Ramanujan's evaluations and more general results by various approaches. The different evaluations arising from different methods yield intriguing, unsuspecting identities.

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained.

-Preface (K. Alladi and F. Garvan).- 1. MacMahon's dream (G. E. Andrews and P. Paule).- 2. Ramanujan's elementary method in partition congruences (B. Berndt, C. Gugg, and S. Kim).- 3. Coefficients of harmonic Maass forms (K. Bringmann and K. Ono).- 4. On the growth of restricted partition functions (E. R. Canfield and H. Wilf).- 5. On applications of roots of unity to product identities (Z. Cao).- 6. Lecture hall sequences, q-series, and asymmetric partition identities (S. Corteel, C. Savage and A. Sills).- 7. Generalizations of Hutchinson's curve and the Thomae formula (H. Farkas).- 8. On the parity of the Rogers-Ramanujan coefficients (B. Gordon).- 9. A survey of the classical mock theta functions (B. Gordon and R. McIntosh).- 10. An application of the Cauchy-Sylvester theorem on compound determinants to a BC_n Jackson integral (M. Ito and S. Okada).- 11. Multiple generalizations of q-series identities found in Ramanujan's Lost Notebook (Y. Kajihara).- 12. Non-terminating q-Whipple transformations for basic hypergeometric series in U(n) (S. C. Milne and J. W. Newcomb).

In previous work arising from the study of Ramanujan's Lost Notebook, a new Abel type lemma was proved. In this paper, we discuss extensions of this lemma and use it to prove many q-series identities.