Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function

We prove that the Riemann zeta-function ζ(σ. +. it) has no zeros in the region σ≥1-1/(5.573412logL{cyrillic}|t|) for |t|≥2. This represents the largest known zero-free region within the critical strip for 3.06{dot operator}1010<|t|<expL{cyrillic}(10151.5). Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials.