This thesis presents new analytic and computational methods designed to improve current techniques for estimating the Vinogradov integral. The resulting bounds are more precise and yield advances in several of its applications within analytic number theory.

The Vinogradov integral is a fundamental tool in analytic number theory. It plays a central role in understanding properties of the Riemann zeta function $\zeta (s)$ near the line $\mathrm {Re}(s)=1$ , including zero-density estimates and zero-free regions. In addition, the Vinogradov integral has applications to other problems, such as the generalised Dirichlet divisor problem.

The first part of the thesis is devoted to obtaining new explicit bounds for the Vinogradov integral

$$ \begin{align*} J_{s,k}(P)= \int_{[0,1]^k} \bigg| \sum_{1\leq x\leq P} e(\alpha_1x+\cdots+\alpha_kx^k) \bigg|^{2s} \,d\boldsymbol{\alpha}, \quad e(z)=e^{2\pi iz}. \end{align*} $$

We consider explicit estimates of the form

$$ \begin{align*} J_{s,k}(P) \leq D(s,k)P^{2s-k(k+1)/2+\eta(s,k)}. \end{align*} $$

Building on the work of Ford [Reference Ford3] and Preobrazhenskiĭ [Reference Preobrazhenskiĭ6], we improve the exponent $\eta (s,k)$ and the ranges in which useful explicit estimates are available. In particular, for large k, the leading term in the exponent is improved from approximately $ \tfrac 38 k^2 e^{1 / 2-2 s / k^2}=0.618\,k^2e^{-2s/k^2}$ to $({8}/{25}) k^2 e^{0.6494-2 s / k^2}=0.612\cdots k^2e^{-2s/k^2}$ .

We also obtain a more convenient version in which the loss in the exponent is fixed at $0.001k^2$ , with explicitly controlled constants, valid for every $k\geq 129$ .

One of the principal applications of explicit bounds for the Vinogradov integral is the derivation of explicit estimates for Vinogradov-type exponential sums. We prove that, for $N\geq 2$ , $N\leq t$ and

$$ \begin{align*} \lambda=\frac{\log t}{\log N}, \end{align*} $$

one has

$$ \begin{align*} \max_{0<u\leq1}\max_{N<R\leq2N} \bigg| \sum_{N<n\leq R}(n+u)^{-it} \bigg| \leq 8.7979\,N^{1-1/(132.94357\lambda^2)}. \end{align*} $$

This improves the corresponding estimate of Ford [Reference Ford3]. Such exponential-sum estimates play a crucial role in the analysis of $\zeta (s)$ , and improvements in these bounds lead directly to sharper explicit estimates for quantities associated with the Riemann zeta function.

In particular, we obtain

$$ \begin{align*} |\zeta(\sigma+it)| \leq 70.6995\,|t|^{4.43795(1-\sigma)^{3/2}} \log^{2/3}|t|, \quad |t|\geq3,\ \tfrac12\leq\sigma\leq1. \end{align*} $$

This yields improved Korobov–Vinogradov zero-free regions. Uniformly for $|t|\geq 3$ , we obtain

$$ \begin{align*} \zeta(\sigma+it)\neq0 \quad\text{whenever}\, \sigma\geq 1-\frac{1} {53.989(\log|t|)^{2/3}(\log\log|t|)^{1/3}}, \end{align*} $$

while, for sufficiently large $|t|$ , the constant $53.989$ may be replaced by $48.0718$ . The latter improves the previous asymptotic constant $48.1588$ of Mossinghoff et al. [Reference Mossinghoff, Trudgian and Yang5].

Another major application concerns explicit zero-density estimates for $\zeta (s)$ . Let

$$ \begin{align*} N(\sigma,T) = \#\{\rho=\beta+i\gamma: \zeta(\rho)=0,\ 0<\gamma<T,\ \beta>\sigma\}. \end{align*} $$

We obtain the fully explicit log-free estimate

$$ \begin{align*} N(\sigma,T) \leq 7.3\cdot10^9T^{11(1-\sigma)} \end{align*} $$

for every $T\geq 3$ and $\sigma \in [0.985,1]$ . This refines earlier explicit zero-density estimates near the line $\sigma =1$ [Reference Bellotti1].

We then develop a different type of zero-density estimate specifically for $\sigma $ close to the left edge of the Korobov–Vinogradov zero-free region. The improved exponential-sum estimates obtained from the Vinogradov integral are combined with a mollifier built from Barban–Vehov weights. If $A>1/48.0718$ is fixed and T is sufficiently large, we prove that

$$ \begin{align*} \sigma\geq 1-\frac{A}{(\log T)^{2/3}(\log\log T)^{1/3}} \quad\Longrightarrow\quad N(\sigma,T)\ll e^{55A}. \end{align*} $$

In particular, the number of zeros is bounded by an absolute constant when $\sigma $ lies sufficiently close to the left edge of the Korobov–Vinogradov zero-free region.

As a consequence, we obtain an improved error estimate in the prime number theorem, given the presently known zero-free regions for $\zeta (s)$ . Writing

$$ \begin{align*} \psi(x)=\sum_{p^m\leq x}\log p, \quad \Delta(x)=\frac{|\psi(x)-x|}{x}, \end{align*} $$

and assuming a Korobov–Vinogradov zero-free region with constant $A_0$ , we prove

$$ \begin{align*} \Delta(x) \ll \exp(55A_0)\exp\{-\omega(x)\}, \end{align*} $$

where

$$ \begin{align*} \omega(x)= \bigg(\frac{5^6A_0^3}{2^2\cdot3^4}\bigg)^{1/5} \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}. \end{align*} $$

This removes the usual $1-\varepsilon $ loss in the exponent appearing in earlier estimates of this type.

Finally, the thesis applies optimal nonexplicit bounds for the Vinogradov integral to the generalised Dirichlet divisor problem. Writing

$$ \begin{align*} \sum_{n\leq x}d_k(n) = xP_{k-1}(\log x)+\Delta_k(x), \end{align*} $$

Heath-Brown [Reference Heath-Brown4] proved, using the modern Vinogradov mean value theorem, that

$$ \begin{align*} \Delta_k(x)\ll x^{1-0.849k^{-2/3}}. \end{align*} $$

In joint work with Yang [Reference Bellotti and Yang2], we develop a method that applies the corresponding exponential-sum estimates directly and obtain

$$ \begin{align*} \Delta_k(x)\ll x^{1-1.889k^{-2/3}} \end{align*} $$

for all sufficiently large fixed k.