This thesis presents new explicit results on the distribution of prime numbers. The results largely fall into the categories of error estimates for the prime number theorem (PNT) and interval estimates for primes. The error in the PNT can be estimated with the truncated Riemann–von Mangoldt explicit formula
$$ \begin{align*}\psi(x) = x- \sum_{|\gamma|\le T} \frac{x^\rho}{\rho} + E(x,T),\end{align*} $$
where
$\rho = \beta +i\gamma $
represents the nontrivial zeros of the Riemann zeta-function. A new explicit version of Goldston’s estimate for
$E(x,T)$
is proved, of order
$$ \begin{align*}E(x,T) = O\bigg(\frac{x\log x\log\log x}{T}\bigg).\end{align*} $$
This estimate is used to update two short-interval results: we prove that there are primes between cubes, that is, in intervals
$(n^3, (n + 1)^3)$
for all
$n \ge \exp (\exp (32.537))$
, and primes between
$n^{155}$
and
$(n + 1)^{155}$
for all
$n \ge 1$
. These results are published in [Reference Cully-Hugill1]. The proof follows the original method of Ingham and builds on work of Dudek [Reference Dudek6]. We also use, and prove, updated versions of Bertrand’s postulate of primes in
$(n, 2n - 2)$
for integers
$n> 3$
. This work is published in [Reference Cully-Hugill and Lee4], with corrections in [Reference Cully-Hugill and Lee5]. The methods of Ramaré and Saouter [Reference Ramaré and Saouter8] and Kadiri and Lumley [Reference Kadiri and Lumley7] are used to give new pairs
$(\Delta , x_0)$
for which there exist at least one prime in
$((1 - \Delta ^{-1})x, x]$
for all
$x \ge x_0$
. For instance, we can take
$(x_0, \Delta ) = (e^{150}, 2.07 \times 10^{11})$
. Lastly, new conditional results are proved for the error in the PNT. Under the Riemann hypothesis (RH), we prove an explicit error estimate and an explicit mean-value estimate for the PNT in short intervals. The former is published in [Reference Cully-Hugill and Dudek2] and the latter is in the preprint [Reference Cully-Hugill and Dudek3]. The mean-value estimate is based on Selberg’s work [Reference Selberg9], and is of particular interest for its applications, of which two are given. We first prove that under the RH, there is a prime in
$(y,y + 37\log ^2 y]$
for at least half the
$y \in [x,2x]$
and all
$x \ge 2$
. The second application is to Goldbach numbers: we prove that under the RH, there is a Goldbach number in
$(x, x + 864 \log ^2 x]$
for all
$x \ge 2$
.
