The Vinogradov Integral and Its Applications

The Vinogradov integral is a fundamental tool in analytic number theory. It plays a central role in understanding properties of the Riemann zeta function ζ(s) near the line Re(s)=1, including zero-density estimates and zero-free regions. In addition, the Vinogradov integral has been applied to diverse problems, such as the generalised Dirichlet divisor problem. 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 its various applications within analytic number theory. One of the principal applications of explicit bounds for the Vinogradov integral is the derivation of explicit estimates for Vinogradov-type exponential sums. Such estimates play a crucial role in the analysis of ζ(s), and any improvement in these bounds leads directly to sharper results for fundamental quantities associated with the Riemann zeta function. In particular, they yield strengthened explicit bounds for ζ(s) in the critical strip and refinements of known zero-free regions. Another major application concerns explicit zero-density estimates for ζ(s). New fully explicit bounds are obtained, including estimates that are effective for values of the real part σ of s close to 1. Furthermore, the improved exponential sum estimates derived from the new bounds for the Vinogradov integral are combined with a mollifier built from Barban–Vehov weights, leading to a new type of zero-density estimate for σ close to 1. More precisely, we show that the number of zeros N(σ,T) can be bounded by an absolute constant when σ lies sufficiently close to the left edge of the Korobov–Vinogradov zero-free region. As a consequence, one obtains an improved error estimate in the prime number theorem, given our current knowledge of zero-free regions for ζ(s). Finally, the thesis applies known optimal non-explicit bounds for the Vinogradov integral to the generalised Dirichlet divisor problem, obtaining improved estimates for the associated error term.