8 papers
An update on the Linnik--Goldbach problem
Daniel R. Johnston, Tim Trudgian
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…
The infinitude of square-free palindromes
Daniel R. Johnston, Bryce Kerr
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…
A round of Pintz to celebrate oscillations in sums
Daniel R. Johnston, Tim Trudgian
We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for -functions. In particular, we make expli…
An effective Bombieri-Vinogradov error term for sifting problems
Daniel R. Johnston
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 o…
Defining newforms in characteristic
Daniel R. Johnston
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 the…
Zero-density estimates and the optimality of the error term in the prime number theorem
Daniel R. Johnston
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…
New bounds and progress towards a conjecture on the summatory function of
Daniel R. Johnston, Nicol Leong, Sebastian Tudzi
In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} where counts the number of prime factors of , with mult…
An explicit version of Chen's theorem and the linear sieve
Matteo Bordignon, Daniel R. Johnston, Valeriia Starichkova
Drawing inspiration from the work of Nathanson and Yamada we prove that every even integer larger than can be written as the sum of a prime and the product of…