arXiv subjects
Publications and source records attributed to Michaela Cully-Hugill.
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 powers.
We provide explicit upper bounds of the order $\log t/\log\log t$ for $|ζ'(s)/ζ(s)|$ and $|1/ζ(s)|$ when $σ$ is close to $1$. These improve existing bounds for $ζ(s)$ on the $1$-line.
We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.
This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two primes) in short intervals. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32277\log^2 y]$ for at least half the $y\in[x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log^2 x]$ for all $x\geq 2$.
This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between $n^{155}$ and $(n+1)^{155}$ for all $n\geq 1$. This result is in part obtained with a new explicit version of Goldston's 1983 estimate for the error in the truncated Riemann--von Mangoldt explicit formula.
Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(Δ,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - Δ^{-1}), x]$.
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number of divisors of $n$ and $d_4(n)$ is the number of ways of writing $n$ as a product of four numbers. In doing so we make a slight improvement on the upper bound for class numbers of quartic number fields.