Properties of reciprocity formulas for the Rogers–Ramanujan continued fractions

The Ramanujan Journal - Ramanujan recorded four reciprocity formulas for the Rogers–Ramanujan continued fractions. Two reciprocity formulas each are also associated with the...     

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...     

Arithmetical properties of multiple Ramanujan sums

In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D.R. Anderson and T.M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.     

A pair of equations in four prime squares and powers of 2

The Ramanujan Journal - In this paper, we obtain when $$k=98$$ , every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of four prime...     

Classical approach to Ramanujan’s modular equations of septic degree

The Ramanujan Journal - In this paper, we prove six Ramanujan’s modular equations of septic degree by employing Ramanujan’s $$_1\psi _1$$ summation formula and certain theta function...     

Modular equations for cubes of the Rogers–Ramanujan and Ramanujan–Göllnitz–Gordon functions and their associated continued fractions

In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions.     

2-Universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms.     

Ramanujan’s partial theta series and parity in partitions

A partial theta series identity from Ramanujan’s lost notebook has a connection with some parity problems in partitions studied by Andrews in Ramanujan J., to appear  where 15 open problems are listed. In this paper, the partial theta series identity of Ramanujan is revisited and answers to Questions 9 and 10 of Andrews are provided.     

Some Identities Involving Certain Hardy Sums and Ramanujan Sum

The main purpose of this paper is, using the mean-value theorem of Dirichlet 1-functions, to study the distribution properties of the hybrid mean value involving certain Hardy sums and Ramanujan sum, and give four interesting identities.     

Concerning an infinite series of Ramanujan related to the natural logarithm

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.     

Cranks—really, the final problem

A survey of Ramanujan’s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks—not mock theta functions.     

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Tóth.     

Srinivasa Ramanujan (1887‐1920) A centennial tribute

This centennial tribute commemorates Ramanujan the man and Ramanujan the mathematician. A brief account of his remarkable mathematical contributions is presented to describe the great talent of Srinivasa Ramanujan. Considerable discussion is made of his new and important ideas, results and theorems in various subjects including definite integrals, hypergeometric series, Bernoulli and Euler numbers, the Riemann zeta function, distribution of prime numbers, theory of partitions and statistical mechanics, continued fractions, hypergeometric functions, Rogers‐Ramanujan identities, theory of representation of numbers as sums of squares, Ramanujan's τ‐function, q‐series, elliptic and modular functions.     

Corrigendum to “on the restricted partition function”

The Ramanujan Journal - This article is a corrigendum to our paper (Cimpoea? and Nicolae in Ramanujan J 473:565–588, 2018).     

Shifted versions of the Bailey and Well-Poised Bailey lemmas

The Bailey lemma is a famous tool to prove Rogers–Ramanujan type identities. We use shifted versions of the Bailey lemma to derive m-versions of multisum Rogers–Ramanujan type identities. We also apply this method to the Well-Poised Bailey lemma and obtain a new extension of the Rogers–Ramanujan identities.     

Ramanujan’s Master Theorem

S. Ramanujan introduced a technique, known as Ramanujan??s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan??s Master Theorem is discussed.     

Geometric deduction of the solutions to modular equations

The Ramanujan Journal - In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were...     

On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function

The Ramanujan Journal - We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the...     

A generalization of Knopp’s Observation on Ramanujan’s tau-function

The Ramanujan Journal - We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.