On an asymptotic series of Ramanujan

An asymptotic series in Ramanujan’s second notebook (Entry 10, Chap. 3) is concerned with the behavior of the expected value of φ(X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ(X) when the distribution of X belongs to a suitable family indexed by a convolution parameter. Examples include the binomial, negative binomial, and gamma families. Some formulas associated with the negative binomial appear new.     

On Ramanujan relations between Eisenstein series

The Ramanujan relations between Eisenstein series can be interpreted as an ordinary differential equation in a parameter space of a family of elliptic curves. Such an ordinary differential equation is inverse to the Gauss–Manin connection of the corresponding period map constructed by elliptic integrals of first and second kind. In this article we consider a slight modification of elliptic integrals by allowing non-algebraic integrands and we get in a natural way generalizations of Ramanujan relations between Eisenstein series.     

Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

Ramanujan series for Epstein zeta functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions \(E^{\varGamma _0(N)}(z,s)\) on the Hecke congruence groups \( \varGamma _0(N),N\in \mathbb {Z}_{>0}\), where z is an arbitrary point in the upper half-plane \( \mathfrak {H}\) and \(s\in \mathbb {Z}_{>1}\). These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1, 2014).     

Some series related to infinite series given by Ramanujan

In this paper we discuss some formulas concerning the summation of certain infinite series, given by Ramanujan in his notebooks [1], vol. 1, Ch. XVI (pp. 251–263), and vol. 2, Ch. XV (pp. 181–192). (A large part of the material in Ch. XVI is contained also in Ch. XV, with only minor changes.) It is shown that several of the formulas given are erroneous. Most of the remaining formulas have by now been proved by residuum calculus. Some of these proofs are extendable to cases which do not seem to have attracted attention earlier. As an example of this we mention the sums $$\sum\limits_{n = 1}^\infty {( - 1)^{n - 1} n^s } /\sinh n\pi = 0 for s = 5,9,13,17,...$$      

Ramanujan type congruences for the Klingen-Eisenstein series

In the case of Siegel modular forms of degree \(n\) , we prove that, for almost all prime ideals \(\mathfrak {p}\) in any ring of algebraic integers, mod \(\mathfrak {p}^m\) cusp forms are congruent to true cusp forms of the same weight. As an application we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan’s congruence. We will conclude by giving numerical examples.     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

Two remarkable doubly exponential series transformations of Ramanujan

The purpose of this note is to prove two doubly exponential series transformations found in Ramanujan’s second notebook. Dedicated to the memory of Professor K G Ramanathan     

On an Identity of Ramanujan

In this paper our aim is to determine all the solutions of the functional equation f(a + b + c) + f(b + c + d) + f(a - d) = f(a + b + d) + f(a + c + d) + f(b - c), where a, b, c, d Zsatisfy ad = bc. This equation is a generalization of one of the identities of Ramanujan. He found two solutions, f(x) = x2, and f(x) = x4. We prove that every solution of the equation can be written as a linear combination of 11 independent solutions.     

On Two Identities of Ramanujan

We use the Ramanujan operator and some modular relations of degree 5 to give new proofs of two identities of Ramanujan.     

On an identity of Ramanujan

An identity stated and utilized by Ramanujan in his now classic study of certain arithmetical functions is proved.

     

Hadamard products for generalized Rogers–Ramanujan series

The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.     

On three identities of Ramanujan

The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to...