Dept of Mathematics and Statistics, Lancaster University

My departmental page (with contact information)

- Banach spaces and operator ideals
- Inequalities and special functions
- (very) analytic number theory

Book "The Prime Number Theorem" Cambridge University Press 2003

The majorisation principle for convex functions

Two ways to generate monotonic sequences: convexity and ratios

Functions with convex derivative;monotonicity of the mid-point and trapezium estimates for integrals

Inequalities for sums of powers: some results of Grahame Bennett

Meaningful and co-meaningful sequences

Hardy's inequality for averages

The norm of a weighted mean operator: Cartlidge's theorem

How close is the approximation by Bernstein polynomials? (updated 2 August 2019)

Approximation to Lipschitz and continuous functions by trigonometric polynomials; Jackson's theorem

Hilbert's inequality and related results

The denominators of the Bernoulli numbers

Binomial power sums and Stirling numbers

A note on series of positive and negative terms

Half-integer estimates for harmonic sums and the digamma function: de Temple's method

Interpolating polynomials and divided differences

Counting zeros of generalized polynomials: Descartes' rule of signs and Laguerre's extensions

Using double integrals to solve single integrals

Integrals evaluated in terms of Catalan's constant (with Nick Lord)

Three answers to an integral (with Nicholas Jameson, Math. Gazette 2006)

Four methods for a trigonometric integral (with Timothy Jameson, Math. Gazette 2013)

The real exponential integrals

The incomplete gamma functions

Integrals of the form $\int_x^\infty f(t)e^{it}\: dt$

The integral $\int_x^\infty \frac{e^{it}}{t^p}\: dt$; Fresnel-type integrals

An inequality for Si$(x)$, (Math. Gazette}, 2015, with Nick Lord and James McKee)

An approximation to the arithmetic-geometric mean (Math. Gazette, 2014)

Elliptic integrals, the arithmetic-geometric mean and the Brent-Salamin algorithm for π

Inequalities for the perimeter of an ellipse (Math. Gazette, 2014)

Basic theory of the gamma function derived from Euler's limit definition

A fresh look at Euler's limit formula for the gamma funnction (Math. Gazette, 2014)

A short proof of the identity linking the beta and gamma integrals

Inequalities for gamma function ratios (American Math. Monthly}, 2013)

A simple proof of Stirling's formula for the gamma function (Math. Gazette}, 2015)

Some results on $\Gamma (1/x)$ and $\psi (1/x)$

Series involving $\zeta (n)$ (Math.\ Gazette, 2014)

Some inequalities for $(a+b)^p$ and $(a+b)^p + (a-b)^p$ (Math. Gazette, 2014)

Inequalities comparing $(a+b)^p - a^p - b^p$ and $a^{p-1}b + ab^{p-1}$

Notes on Fermat's Last Theorem

Two squares and four squares: the simplest proof of all?

Stan Dolan's proof of the two-squares theorem

Revisiting even and odd square-free numbers

Sums and products of algebraic numbers

Counting divisors (Math. Gazette, 2014)

Expressing harmonic sums as fractions (Math. Gazette 101 (2017))

Carmichael numbers with three prime factors

Carmichael numbers and pseudoprimes

An inequality for 3-factor Carmichael numbers due to J.M. Chick

Goats and birds (Math. Gazette 101 (2017), with Nicholas Jameson)

The fallacy in the attempted proof of the Riemann hypothesis by Liu, Rybachuk and Liu

My son Tim (alias Timothy or T.P.) died tragically on 5 September 2013. He was passionately keen on mathematics, though he did not complete a Ph. D. or hold a paid position.

The following is a selection of his mathematical writings, both published and unpublished (in addition to some written jointly with myself, listed above) , The unpublished ones were generally notes intended only for his own use, but some of them may be of interest to a wider audience. Most of them lacked a list of references which I have endeavoured to supply.

The probability integral by volume of revolution (Math. Gazette, 1994)

Another proof that zeta(2) = pi^2/6 by double integration (Math Gazette, 2013)

Orbits made easy by complex numbers (Math. Gazette, 2014)

Some double series related to zeta(3) (Math. Gazette, 2014)

Polylogarithms, multiple zeta values, and the series of Hjortnaes and Comtet

An elementary derivation of the Poisson summation formula

The fourth power moment of zeta(1/2 + it)

The approximate functional equation for zeta(s)^2

The Ingham-Huxley zero-density theorem

Linnik's proof of the Waring-Hilbert theorem from Hua's book (with a mistake corrected)

The Gauss circle problem (the classical exponent 1/3)

Integrating the error in the divisor problem

Daboussi's version of Vinogradov's bound for the exponential sum over primes

Member of Lancaster Baptist Church and church representative for Christian Aid

Mountain walking (not hard climbing!)

Member of Austrian
Alpine Club

Articles on mountain walking:

- The Japan Alps (Austrian Alpine Club Newsletter 73, 1982)
- Jotunheimen (Austrian Alpine Club Newsletter 97, 1988)
- Introducing the Children to the Alps (1989)
- The Schladminger Tauern (Austrian Alpine Club Newsletter 130, 1996), revised and expanded 2002
- The Lochaber Traverse (1996)
- Kilimanjaro (2003)

Scottish dancing

instructor of class at Hest
Bank (Tuesdays 7.30 p.m.)

holder of teaching certificate
of the Royal Scottish Country Dance Society

Running

best marathon time 2:54:37
(Wolverhampton, 1983 - now a long time ago!)

Election results under the d'Hondt method;

- The d'Hondt Electoral System
- UK European Parliament Election Results, May 2014
- Scottish Parliament Election 2016
- Welsh Assembly Election 2016

How old am I? (simple number games)