For $r\in(0,1)$, let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$, where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that

\begin{equation*}\mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{2n}r^{2n},\end{equation*}

with $\theta _{n}\in \left( 0,1\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

Let $\Omega $ be a complex lattice which does not have complex multiplication and $\wp =\wp _\Omega $ the Weierstrass $\wp $-function associated with it. Let $D\subseteq \mathbb {C}$ be a disc and $I\subseteq \mathbb {R}$ be a bounded closed interval such that $I\cap \Omega =\varnothing $. Let $f:D\rightarrow \mathbb {C}$ be a function definable in $(\overline {\mathbb {R}},\wp |_I)$. We show that if f is holomorphic on D then f is definable in $\overline {\mathbb {R}}$. The proof of this result is an adaptation of the proof of Bianconi for the $\mathbb {R}_{\exp }$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.

The signature four elliptic theory of Ramanujan is provided with a counterpart to the Jacobian modular sine; this counterpart yields natural direct proofs of several hypergeometric identities recorded by Ramanujan, bypassing the signature four transfer principle of Berndt et al. [‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc. 347 (1995), 4163–4244].

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ‘arithmetic–geometric mean (AGM)’ integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_{0}$, we call the resulting series ‘Ramanujan series with the shift $x_{0}$’. Then we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell =1,2,3$ and the shift $x_{0}=1/2$.

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential equations satisfied by the sigma-function. By way of application, we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schrödinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank 3 lattice in ℂ2 are given explicitly.

We prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$

$k=k(q)$

$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

We prove that there is a correspondence between Ramanujan-type formulas for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet $L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

We consider the classical problem of finding the best uniform approximation by polynomials of $1/(x-a)^2,$ where $a>1$ is given, on the interval $[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

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.

Generating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

The statement of the numerical values $\kappa$ and $z_0$ on page 167 of [1, Section 3] contained an error. The values that were actually used were (to nine decimal places): $$\begin{eqnarray*} \kappa &=& 2 \omega_1 -\int_1^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&1.859\,185\,431, \\[6pt] z_0&=&2\omega_3+\int_{-1}^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&0.204\,680\,500+1.225\,694\,691i, \end{eqnarray*}$$ these being shifted, by the periods $2\omega_1$ and $2\omega_3$ respectively, compared with the values given in [1] (with $\omega_1=1.496\,729\,323$ and $\omega_3=1.225\,694\,691i$). With $\tau_0=\tau_1 =1$ and $\sigma (z)$ denoting the sigma function $\sigma (z;g_2,g_3)$ with invariants $g_2=4$, $g_3=-1$ associated with the elliptic curve given by equation (3.2), these values of $\kappa$ and $z_0$ yield $$\begin{eqnarray*} \sigma (\kappa )&=&1.555\,836\,426, \\[4pt] A=\frac{\tau_0}{\si (z_0 )} &=& 0.112\,724\,016-0.824\,911\,686i, \\[4pt] B=\frac{\si (\ka )\si (z_0 )\,\tau_1} {\si (z_0+\ka )\,\tau_0}&=&0.215\,971\,963+0.616\,028\,193i. \end{eqnarray*}$$ and the latter three values all agree with those stated in the paper (apart from rounding down the last digit in the imaginary part of $A$).

The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus.

A new infinite product ${{t}_{n}}$ was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about ${{t}_{n}}$ by establishing new connections between the modular $j$-invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, ${{t}_{n}}$ generates the Hilbert class field of $\mathbb{Q}\left( \sqrt{-n} \right)$. This shows that ${{t}_{n}}$ is a new class invariant according to H. Weber’s definition of class invariants.