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A. J. Irving

Publications and source records attributed to A. J. Irving.

9 recordsLinked to original sources
arXiv2015

Cubic polynomials represented by norm forms

We show that for an irreducible cubic $f\in\mathbb Z[x]$ and a full norm form $\mathbf N(x_1,\ldots,x_k)$ for a number field $K/\mathbb Q$ satisfying certain hypotheses the variety $f(t)=\mathbf N(x_1,\ldots,x_k)\ne 0$ satisfies the Hasse principle. Our proof uses sieve methods.

math.NT↗
arXiv2014

The largest prime factor of $X^3+2$

Improving on a theorem of Heath-Brown, we show that if $X$ is sufficiently large then a positive proportion of the values $\{n^3+2:n\in (X,2X]\}$ have a prime factor larger than $X^{1+10^{-52}}$.

math.NT↗
arXiv2014

Almost-prime values of polynomials at prime arguments

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the degree is large, there are infinitely many primes $p$ for which $f(p)$ has at most $°f+O(\log°f)$ prime factors.

math.NT↗
arXiv2014

The divisor function in arithmetic progressions to smooth moduli

By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.

math.NT↗
arXiv2013

Diophantine Approximation with Products of Two Primes

We show that for any irrational $α$ and any $τ<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|nα\|\leq n^{-τ}.$$ We also show that for all sufficiently large $b$ there exist 3-digit palindromes in base $b$ with precisely two prime factors.

math.NT↗