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math.NT2024
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…
math.NT2024
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…
math.NT2024
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…