@Article{Zaccagnini2000b, Author = "A. Zaccagnini", Title = "A conditional density theorem for the zeros of the {R}iemann zeta-function", Journal = "Acta Arith.", Volume = "93.3", Pages = "293--301", Year = 2000, Abstract = "We find zero-free regions and density theorems for the Riemann zeta-function depending on non-trivial estimates for $$ J(x,\theta) := \int_x^{2x} | \psi(t) -\psi(t-\theta t) - \theta t |^2 dt, $$ uniformly in some range of $\theta$. A Corollary of our main result is the following: Assume that $J(x,\theta) = \mathcal{O}\left( x^3 \theta^2 (F(x\theta))^{-1} \right)$, uniformly for $x^{-\beta}\le \theta\le 1$, where $F$ is a positive, increasing function which is unbounded as $x\to+\infty$, such that $F(x) = \mathcal{O}(x^\epsilon)$ for every $\epsilon>0$, and that $\beta\in(0,1)$. Then the Riemann zeta-function does not vanish in the region $$ \sigma > 1- C {\log F(t)\over \log t}, $$ where $C$ is some absolute, positive constant. The proof is based on Tur{\'a}n's power sum method." }