One can use an argument based on work by Vinogradov to show that the Riemann zeta function has a zero-free region very close to the line $\text{Re }s = 1$. Titchmarsh describes this approach in chapter 6 of his book on the zeta function and also notes that Karatsuba presents a greatly simplified argument using $p$-adic methods.
This latter material can be found in Karatsuba's book on analytic number theory, but I do not have access to that volume. Is there a source online which describes this approach?