In his “lost” notebook, Ramanujan stated two results, which are equivalent to the identities

$$\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{n}} \right)}^{5}}}{\left( 1-{{q}^{5n}} \right)}=1-5\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{d} \right)d \right){{q}^{n}}$$

and

$$q\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{5n}} \right)}^{5}}}{\left( 1-{{q}^{n}} \right)}=\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{n/d} \right)d \right){{q}^{n}}.$$

We give several more identities of this type.

The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of ${{\mathbb{Z}}_{4}}$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

In this paper, we give the ${{L}^{p}}$ boundedness for a class of parabolic Littlewood–Paley $g$-function with its kernel function $\Omega$ is in the Hardy space ${{H}^{1}}\left( {{S}^{n-1}} \right)$.

We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.

This paper introduces the concepts of fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity in intuitionistic fuzzy topological spaces. After defining the fundamental concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces, we present intuitionistic fuzzy $\gamma$-open sets and intuitionistic fuzzy $\gamma$-continuity and other results related topological concepts.

This paper is concerned with the boundary behavior of solutions of the Helmholtz equation in ${{\mathbb{R}}^{n}}$. In particular, we give a Littlewood-type theorem to show that the approach region introduced by Korányi and Taylor (1983) is best possible.

A ring $R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to ${{C}^{*}}$-algebras. Various examples to illustrate and delimit our results are provided.

We compute the minimal polynomials of the Ramanujan values ${{t}_{n}}$, where $n\,\equiv \,11\,\bmod \,24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}\left( \sqrt{-n} \right)$ and have much smaller coefficients than the Hilbert polynomials.

We say that a numerical semigroup is $d$-squashed if it can be written in the form

$$S\,=\,\frac{1}{N}\langle {{a}_{1}},\,.\,.\,.\,,\,{{a}_{d}}\rangle \,\cap \,\mathbb{Z}$$

for $N$, ${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$ positive integers with $\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular.

Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture.

We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

Let $f\in \,{{L}_{2\pi }}$ be a real-valued even function with its Fourier series $\frac{{{a}_{0}}}{2}\,+\,\sum _{n=1}^{\infty }\,{{a}_{n}}\,\cos \,nx$, and let ${{S}_{n}}\left( f,x \right)$, $n\,\,\ge \,\,1$, be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{{{a}_{n}}\}$ is decreasing and ${{\lim }_{n\to \infty }}\,{{a}_{n}}\,=\,0$, then

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left\| f-{{S}_{n}}\left( f \right) \right\|}_{L}}=0\text{ifanyonlyif}\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\log n=0.$$

We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the ${{L}^{1}}$-convergence of a function $f\in {{L}_{2\pi }}$ in complex space. We also give results on ${{L}^{1}}$-approximation of a function $f\in {{L}_{2\pi }}$ under the MVBV condition.