Zobrazeno 1 - 10
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pro vyhledávání: '"Johnston, P. R."'
In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(
Externí odkaz:
http://arxiv.org/abs/2408.04143
We give an explicit $O(x/T)$ error term for the truncated Riemann--von Mangoldt explicit formula. For large $x$, this provides a modest improvement over previous work, which we demonstrate via an application to a result on primes between consecutive
Externí odkaz:
http://arxiv.org/abs/2402.04272
Autor:
Sharma, Mrinank, Tong, Meg, Korbak, Tomasz, Duvenaud, David, Askell, Amanda, Bowman, Samuel R., Cheng, Newton, Durmus, Esin, Hatfield-Dodds, Zac, Johnston, Scott R., Kravec, Shauna, Maxwell, Timothy, McCandlish, Sam, Ndousse, Kamal, Rausch, Oliver, Schiefer, Nicholas, Yan, Da, Zhang, Miranda, Perez, Ethan
Human feedback is commonly utilized to finetune AI assistants. But human feedback may also encourage model responses that match user beliefs over truthful ones, a behaviour known as sycophancy. We investigate the prevalence of sycophancy in models wh
Externí odkaz:
http://arxiv.org/abs/2310.13548
We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq 2\cdot 10
Externí odkaz:
http://arxiv.org/abs/2303.13785
Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 369 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this re
Externí odkaz:
http://arxiv.org/abs/2208.01229
Drawing inspiration from the work of Nathanson and Yamada we prove that every even integer larger than $\exp (\exp (32.7))$ can be written as the sum of a prime and the product of at most two primes.
Externí odkaz:
http://arxiv.org/abs/2207.09452
Autor:
Johnston, Daniel R., Males, Joshua
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 $\eta$-functions. By utilising known properties
Externí odkaz:
http://arxiv.org/abs/2204.05465
Autor:
Johnston, Daniel R., Yang, Andrew
By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all $x\geq 2$ that
Externí odkaz:
http://arxiv.org/abs/2204.01980
Autor:
Johnston, Daniel R.
We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(\pi(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This makes expl
Externí odkaz:
http://arxiv.org/abs/2201.06184
We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime numb
Externí odkaz:
http://arxiv.org/abs/2111.10001