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

Abstract

We prove that the Riemann zeta-function $\zeta (\sigma +it)$ has no zeros in the region $\sigma \ge 1-1/(5.573412\log |t|)$ for $|t|\ge 2$. This represents the largest known zero-free region within the critical strip for $3.06\cdot 10^{10}<|t|<\exp (10151.5)$. 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.

17 pages, 2 figures, 5 tables