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Daniel R. Johnston

Publications and source records attributed to Daniel R. Johnston.

At least 19 recordsLinked to original sources
arXiv2026

An effective Bombieri-Vinogradov error term for sifting problems

In number theory, many major results related to the additive properties of primes are proven using the methods of sieve theory. However, in nearly every case, the existing proofs of these results are ineffective, in that explicit values for which they hold cannot be computed. The reason for this ineffectivity is due to the reliance on the Bombieri--Vinogradov theorem. In this paper, we show that any classical sifting problem with a Bombieri--Vinogradov style error term can in fact be made effective, with no loss to the asymptotic form of the original (ineffective) result. This is done by carefully modifying the sieve upper and lower bounds as to avoid the usual complications regarding the existence of a Siegel zero. We also provide some simple applications. For example, we show that one may effectively bound the number of primes $p\leq x$ such that $p+2$ is also prime by \begin{equation*}(4+o(1))C_2\frac{x}{(\log x)^2},\end{equation*}where\begin{equation*}C_2=2\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right).\end{equation*}

math.NT↗
arXiv2026

A round of Pintz to celebrate oscillations in sums

We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of $L$-functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the Möbius function, but we also outline the utility of this method in general.

math.NT↗
arXiv2026

An update on the Linnik--Goldbach problem

We consider the Linnik--Goldbach problem of writing all large even integers as the sum of two primes and a fixed number of powers of 2. We show that, under the generalised Riemann hypothesis, one can use 6 powers of two. In addition, we discuss refinements to the unconditional case and to the related problem of Romanov in expressing a positive proportion of odd numbers as the sum of a prime and a power of 2.

math.NT↗
arXiv2026

The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+η$, $μ^2(η)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length, unlike previous versions.

math.NT↗
arXiv2026

Primes and almost primes between cubes

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition, we use this computation and a sieve-theoretic argument to show that there exists a number with at most 2 prime factors (counting multiplicity) between $n^3$ and $(n+1)^3$ for all $n\geq 1$. Our sieving argument uses a logarithmic weighting procedure attributed to Richert, which yields significant numerical improvements over previous approaches.

math.NT↗
arXiv2026

The infinitude of square-free palindromes

We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski.

math.NT↗
arXiv2025

Power-free palindromes and reversed primes

We prove new results related to the digital reverse $\overleftarrow{n}$ of a positive integer $n$ in a fixed base $b$. First we show that for $b\geq 26000$, there exists infinitely many primes $p$ such that $\overleftarrow{p}$ is square-free. Further, we show that for $b\geq 2$ there are infinitely many palindromes (with $n=\overleftarrow{n}$) that are cube-free. We also give asymptotic expressions for the counting functions corresponding to these results. The main tools we use are recent bounds from the literature on reversed primes and palindromes in arithmetic progressions.

math.NT↗
arXiv2025

Zero-density estimates and the optimality of the error term in the prime number theorem

We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term for some choices of the zero-free region. As an example, we show that if there are no zeros $ρ=β+it$ of $ζ(s)$ with \begin{equation*} 1-β<\frac{1}{c(\log t)^{2/3}(\log\log t)^{1/3}}=:η(t), \end{equation*} then \begin{equation*} \frac{|ψ(x)-x|}{x}\ll\exp(-ω(x))\frac{(\log x)^9}{(\log\log x)^3}, \end{equation*} where $ψ(x)$ is the Chebyshev prime-counting function, and \begin{equation*} ω(x)=\min_{t\geq 3}\{η(t)\log x+\log t\}. \end{equation*} This refines the best known error term for the prime number theorem, previously given by \begin{equation*} \frac{|ψ(x)-x|}{x}\ll_{\varepsilon}\exp(-(1-\varepsilon)ω(x)) \end{equation*} for any $\varepsilon>0$.

math.NT↗
arXiv2025

Evaluating and improving wave and non-wave stress parametrisations for oceanic flows

Whenever oceanic currents flow over rough topography, there is an associated stress that acts to modify the flow. In the deep ocean, this stress is predominantly a form drag due to pressure differentials across topography, caused by the formation of internal waves and other baroclinic motions: processes that act on such small scales most global ocean models cannot resolve. Despite the need to incorporate this stress into ocean models, existing parametrisations are limited in their applicability. For instance, most parametrisations are only suitable for small-scale topography and are either for periodic or steady flows, but rarely a combination thereof. Here we summarise some of the most widely used parametrisations and evaluate the accuracy of a carefully selected subset using hundreds of idealised two-dimensional and three-dimensional simulations spanning a wide parameter space. We focus on the case of an isolated Gaussian hill as an idealised representation of a seamount. In cases where the parametrisations prove to be inaccurate, we use our data to suggest improved formulations. Our results thus provide a starting point for a comprehensive parameterisation of topographic stresses in ocean models where fine scale topography is unresolved.

physics.ao-ph↗
arXiv2025

Almost primes between all squares

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods.

math.NT↗
arXiv2025

The sum of a prime power and an almost prime

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take $M(k)=(2+\varepsilon)k$ for any $\varepsilon>0$, or $M(k)=(1+\varepsilon)k$ under the assumption of the Elliott--Halberstam conjecture. Moreover, we give a variant of this result which accounts for congruence conditions and strengthens a classical theorem of Erdős and Rao. The main tools we employ are the weighted sieve method of Diamond, Halberstam and Richert, bounds on the number of representations of an integer as the sum of two $k$th powers, and results on $k$th power residues. We also use some simple computations and arguments to conjecture an optimal value of $M(k)$, as well as a related variant of Hardy and Littlewood's Conjecture H.

math.NT↗
arXiv2025

Some explicit results on the sum of a prime and an almost prime

Inspired by a classical result of Rényi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 31 prime factors.

math.NT↗
arXiv2024

Defining newforms in characteristic $p$

The theory of newforms, due to Atkin and Lehner, provides a powerful method for decomposing spaces of modular forms. However, many problems occur when trying to generalise this theory to characteristic $p$. Recently, Deo and Medvedovsky have suggested a way around these problems by using purely algebraic notions to define newforms. In this thesis, we describe the methods of Deo and Medvedovsky in detail and generalise their results where possible.

math.NT↗
arXiv2024

New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} where $Ω(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(x)|<2260x$ for all $x\geq 1$. This result provides new progress towards a conjecture of Sun, which asks whether $|W(x)| 0$.

math.NT↗
arXiv2024

On the error term in the explicit formula of Riemann--von Mangoldt

We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.

math.NT↗
arXiv2023

On the sum of a prime and a square-free number with divisibility conditions

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd $k\leq 10^5$ and even $k\leq 2\cdot 10^5$ that any even integer $n\geq 40$ can be expressed as the sum of a prime and a squarefree number coprime to $k$. We also discuss applications to other Goldbach-like problems.

math.NT↗