arXiv subjects

Daniel R. Johnston

Publications and source records attributed to Daniel R. Johnston.

23 records · Page 2Linked to original sources

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.

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arXiv2022

Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $π(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|π(x)-\text{li}(x)|<\sqrt{x}\log x/8π$ for $2657\leq x\leq 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.

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arXiv2022

Some explicit estimates for the error term in the prime number theorem

By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|π(x)-\text{li}(x)|$, $|θ(x)-x|$ and $|ψ(x)-x|$ appearing in the prime number theorem. For example, we show for all $x\geq 2$ that $|ψ(x)-x|\leq 9.39x(\log x)^{1.515}\exp(-0.8274\sqrt{\log x})$. Our estimates rely heavily on explicit zero-free regions and zero-density estimates for the Riemann zeta-function, and improve on existing bounds for prime-counting functions for large values of $x$.

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arXiv2022

Exact formulae and Turán inequalities for Vafa-Witten invariants of $K3$ surfaces

We consider three different families of Vafa-Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa-Witten invariants is given by combinations of twisted Dedekind $η$-functions. By utilising known properties of these $η$-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Turán inequalities.

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arXiv2022

On the average value of $π(t)-\text{li}(t)$

We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(π(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $π(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function $θ(t)$ and discuss the extent to which one can make related claims unconditionally.

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