Thomas Morrill

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20182020

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math.NT2020

Quasimodularity of the $k$ th Residual Cranks

Thomas Morrill, Aleksander Simonič

We establish quasimodularity for a family of residual crank generating functions defined on overpartitions. We also show that the second moments of these $k$ th residual cranks admi…

math.NT2019

Inequalities for the $d$ th Residual Crank Moments of Overpartitions

Ali H. Al-Saedi, Thomas Morrill, Holly Swisher

Two analogues of the crank function are defined for overpartitions -- the first residual crank and the second residual crank. This suggests an exploration of crank functions define…

math.NT2019

Sign changes in the prime number theorem

Thomas Morrill, Dave Platt, Tim Trudgian

Let $V (T)$ denote the number of sign changes in $ψ (x) - x$ for $x \in [1, T]$ . We show that $lim inf_{T \to \infty} V (T) / lo g T \geq γ_{1} / π + 1.867 \cdot 1 0^{- 30}$ , where $γ_{…

math.NT2018

An elementary bound on Siegel zeroes

Thomas Morrill, Tim Trudgian

We consider Dirichlet $L$ -functions $L (s, χ)$ where $χ$ is a real, non-principal character modulo $q$ . Using Pintz's refinement of Page's theorem, we prove that for $q \geq 3$ the f…

math.NT2018

Robin's inequality for 20-free integers

Thomas Morrill, David Platt

In 1984, Robin showed that the Riemann Hypothesis for $ζ$ is equivalent to demonstrating $σ (n) < e^{γ} n lo g lo g n$ for all $n > 5040$ . Robin's inequality has since been proven for…