Vinogradov’s method and density theorems for Riemann’s zeta-function

We prove a series of density theorems for Riemann's zeta-function for the number of zeros lying near to the boundary line $\text{Re}\, s =1$ of the critical strip. In particular, we improve the constant appearing in the exponent of the Halász-Turán density theorem. The proof uses the relatively recent strong estimate for the zeta-function near the line $\text{Re}\, s =1$ showed by Heath-Brown. The necessary exponential sums were estimated by Heath-Brown via the new results of Wooley and of Bourgain, Demeter and Guth on the Vinogradov's mean value integral.