# Volume 42 - Special Commemorative volume in honour of Alan Baker - 2019

### 1. Class Numbers of Quadratic Fields

We present a survey of some recent results regarding the class numbers of quadratic fields

### 2. On the Galois group of Generalised Laguerre polynomials II

For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by$L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}.$These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ when $u$ is a negative integer.

### 3. Linear forms in logarithms and exponential Diophantine equations

This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

### 4. Polynomial representations of GL(m|n)

We develop a modular version of a super analogue of Schur's duality by means of supergroups, rather than Lie superalgebras, in preparation for a geometric analogue.

### 5. Density modulo 1 of a sequence associated to some multiplicative functions evaluated at polynomial arguments

We study density modulo 1 of the sequence with general term ∑ m≤n f (G(m)) where f is the strongly multiplicative function of the form f (n) = ∏ p|n (1 − ν(p) p) and ν is a multiplicative function for which there exists a real number 0 < r ≤ 1 such that 1 ≤ |ν(p)| ≤ p

### 6. Multiplicatively dependent vectors with coordinates algebraic numbers

We shall prove that close to each point in \mathbb{C}^n with coordinates of comparable size there is a point (t_1 , ... , t_n) with the property that no multiplicatively dependent vector (u_1 , ... , u_n) with coordinates which are algebraic numbers of height at most H and degree at most d is very close to (t_1 , ... , t_n).

### 7. A transference inequality for rational approximation to points in geometric progression

We establish a transference inequality conjectured by Badziahin and Bugeaud relating exponents of rational approximation of points in geometric progression.

### 8. Some New Congruences for l-Regular Partitions Modulo 13, 17, and 23

A partition of n is l-regular if none of its parts is divisible by l. Let b l (n) denote the number of l-regular partitions of n. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for b l (n) modulo l, where l = 17, 23, and for b 65 (n) modulo 13.

### 9. Divisibility of Selmer groups and class groups

In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.

### 10. On some Lambert-like series

In this note, we study radial limits of power and Laurent series which are related to the Lerch zeta-function or polylogarithm function. As has been pointed out in [CKK18], there have appeared many instances in which the imaginary part of the Lerch zeta-function was considered by eliminating the real part by considering the odd part only. Mordell studied the properties of the power series resembling Lambert series, and in particular considered whether the limit function is a rational function or not. Our main result is the elucidation of the threshold case of b_n = 1/n studied by Mordell [Mor63], revealing that his result is the odd part of Theorem 1.1 in view of the identities (1.9), (1.5). We also refer to Lambert series considered by Titchmarsh [Tit38] in connection with Estermann's zeta-functions.