Vinogradov’s method

Abstract

Weyl’s method of estimating trigonometric sums was used in the previous chapter to prove Littlewood’s theorem on the zero-free region of ζ (s). Littlewood’s theorem was used, in turn, to obtain the following estimate of the error term in the prime number theorem:

$$\pi (x) - li\,x = O(x\,{e^{ - a\sqrt {\log x\log \log x} }})$$

,for a positive, absolute constant a. A powerful refinement of Weyl’s method was effected by I. M. Vinogradov, who applied it to the solution of a variety of problems in number theory. We shall describe the essentials ofthat method in this chapter, and use it to deduce Chudakov’s refinement of Littlewood’s theorem, to the effect that there exists a constant A₁>0, such that

$$\zeta (s) \ne 0,\,for\quad \sigma \geqslant 1 - \frac{{{A_1}}} {{{{\log }^{\frac{3} {4}}}t{{(\log \log t)}^{\frac{3} {4}}}}},\,t \geqslant {t_1}$$

t≥t₁