Ramanujan’s enigmatic formula for the harmonic series

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.     

Ramanujan — Fourier series and a theorem of Ingham

Given two arithmetical functions f, g, we derive, under suitable conditions, asymptotic formulas for the convolution sums ∑ _n≤N f (n) _g (n + h) for a fixed number h. To this end, we develop the theory of Ramanujan expansions for arithmetical functions. Our results give new proofs of some old results of Ingham proved by him in 1927 using different techniques.     

On Chudnovsky–Ramanujan type formulae

In a well known 1914 paper, Ramanujan gave a number of rapidly converging series for \(1/\pi \) which are derived using modular functions of higher level. Chudnovsky and Chudnovsky in their 1988 paper derived an analogous series representing \(1/\pi \) using the modular function J of level 1, which results in highly convergent series for \(1/\pi \), often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard–Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard–Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant J of level 1, we determine all Chudnovsky–Ramanujan type formulae which are valid around one of the three singular points: \(0, 1, \infty \).     

Bernstein–Sato polynomials of hyperplane arrangements

We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than \(-2\) and the multiplicity of the root \(-1\) is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.     

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.     

Polygonal numbers and Rogers–Ramanujan–Gordon theorem

The Ramanujan Journal - Let $$B_{k,r}(n)$$ be the number of partitions of the form $$n=b_1+b_2+cdots +b_s$$ , where $$b_i-b_{i+k-1}hbox {,char 062,}2$$ and at most $$r-1$$ of the $$b_i$$ are...     

Shift-plethystic trees and Rogers–Ramanujan identities

The Ramanujan Journal - By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the...     

On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrödinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.     

Weighted Rogers–Ramanujan partitions and Dyson crank

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank.     

Explicit Evaluation Formula for Ramanujan’s Singular Moduli and Ramanujan–Selberg Continued Fractions

Mathematical Notes - At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $$alpha_n$$ . All those results were proved by Berndt et. al by using...     

Arc spaces and the Rogers–Ramanujan identities

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations, we obtain a new approach to the classical Rogers–Ramanujan Identities. The linking object is the Hilbert–Poincaré series of the arc space over a point of the base variety. In the case of the double point, this is precisely the generating series for the integer partitions without equal or consecutive parts.     

Odd values of the Rogers–Ramanujan functions

Let $g(n)$ and $h(n)$ be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of $g(n)$ for odd n, and $h(n)$ for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of $g(n)$ for even n, and $h(n)$ for odd n.     

The error term in the Sato–Tate conjecture

Let \({f(z) = \sum_{n=1}^\infty a(n)e^{2\pi i nz} \in S_k^{\mathrm{new}}(\Gamma_0(N))}\) be a newform of even weight \({k \geq 2}\) that does not have complex multiplication. Then \({a(n) \in \mathbb{R}}\) for all n; so for any prime p, there exists \({\theta_p \in [0, \pi]}\) such that \({a(p) = 2p^{(k-1)/2} {\rm cos} (\theta_p)}\) . Let \({\pi(x) = \#\{p \leq x\}}\) . For a given subinterval \({[\alpha, \beta]\subset[0, \pi]}\) , the now-proven Sato–Tate conjecture tells us that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} \sim \mu_{ST} ([\alpha, \beta])\pi(x),\quad \mu_{ST} ([\alpha, \beta]) = \int\limits_{\alpha}^\beta \frac{2}{\pi}{\rm sin}^2(\theta) d\theta. $$ Let \({\epsilon > 0}\) . Assuming that the symmetric power L-functions of f are automorphic, we prove that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} = \mu_{ST} ([\alpha, \beta])\pi(x) + O\left(\frac{x}{(\log x)^{9/8-\epsilon}} \right), $$ where the implied constant is effectively computable and depends only on k,N, and \({\epsilon}\) .     

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.