# Matthew LettingtonCardiff University | CU · School of Mathematics

Matthew Lettington

PhD Number Theory

## About

22

Publications

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Introduction

Additional affiliations

September 2010 - August 2020

## Publications

Publications (22)

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalge...

We identify and analyse obstructions to factorisation of integer matrices into products NTN or N2 of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We...

The jth divisor function \(d_j\), which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function \(c_j\), which counts the ordered factorisations of a positive integer into j factors each of which...

Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural e...

The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive...

An equivalent representation of constant sum matrices in terms of block-structured matrices is given in this paper. This provides an easy way of constructing all constant sum matrices, including those with further symmetry properties. The block representation gives a convenient description of the dihedral equivalence of such matrices. It is also sh...

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an...

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H....

We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and m-dimensional recurrence relations. For each integer m, there exist both rational and integer versions of these sequences, where the underlying prime congruence structures of the rational sequence denominators enables the integer sequence to...

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complement...

Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured matrices. This block representation provides a way of constructing such matrices with further symmetries and of studying their algebraic behaviour, significantly advanci...

Denote by $\Sigma n^m$ the sum of the $m$-th powers of the first $n$ positive
integers $1^m+2^m+\ldots +n^m$. Similarly let $\Sigma^r n^m$ be the $r$-fold
sum of the $m$-th powers of the first $n$ positive integers, defined such that
$\Sigma^0 n^{m}=n^m$, and then recursively by $\Sigma^{r+1} n^{m}=\Sigma^{r}
1^{m}+\Sigma^{r} 2^{m}+\ldots + \Sigma^...

We study higher-dimensional interlacing Fibonacci sequences and their
corresponding multi-dimensional continued fractions, generated via both
Chebyshev type functions and $m$-dimensional recurrence relations. For each
integer $m$, there exist both rational and integer versions of these sequences,
where the underlying $p$-adic structure of the ratio...

The present work concerns the algebra of semi-magic square matrices. These can be decomposed into matrices of specific rotational symmetry types, where the square of a matrix of pure type always has a particular type. We examine the converse problem of categorising the square roots of such matrices, observing that roots of either type occur, but on...

We consider the Mellin transforms of certain Legendre functions based upon
the ordinary and associated Legendre polynomials. We show that the transforms
have polynomial factors whose zeros lie all on the critical line Re $s=1/2$.
The polynomials with zeros only on the critical line are identified in terms of
certain $_3F_2(1)$ hypergeometric functi...

We consider the Mellin transforms of certain Chebyshev functions based upon
the Chebyshev polynomials. We show that the transforms have polynomial factors
whose zeros lie all on the critical line or on the real line. The polynomials
with zeros only on the critical line are identified in terms of certain
$_3F_2(1)$ hypergeometric functions. Furtherm...

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hyp...

We study the interplay between recurrences for zeta related functions at
integer values, `Minor Corner Lattice' Toeplitz determinants and integer
composition based sums. Our investigations touch on functional identities due
to Ramanujan and Grosswald, the transcendence of the zeta function at odd
integer values, the Li Criterion for the Riemann Hyp...

Let the \emph{Fleck numbers}, $C_n(t,q)$, be defined such that
\[
C_n(t,q)=\sum_{k\equiv q (mod n)}(-1)^k\binom{t}{k}.
\]
For prime $p$, Fleck obtained the result $C_p(t,q)\equiv 0 (mod p^{{\left \lfloor (t-1)/(p-1)\right \rfloor}}} )$, where $\lfloor.\rfloor$ denotes the usual floor function. This congruence was extended 64 years later by Weisman,...

## Projects

Project (1)