Explicit zero-free regions for the Riemann zeta-function

Abstract

AbstractWe prove that the Riemann zeta-function $$\zeta (\sigma + it)$$ ζ ( σ + i t ) has no zeros in the region $$\sigma \ge 1 - 1/(55.241(\log |t|)^{2/3} (\log \log |t|)^{1/3})$$ σ ≥ 1 - 1 / ( 55.241 ( log | t | ) 2 / 3 ( log log | t | ) 1 / 3 ) for $$|t|\ge 3$$ | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $$\sigma \ge 1 - 1/(5.558691\log |t|)$$ σ ≥ 1 - 1 / ( 5.558691 log | t | ) for $$|t|\ge 2$$ | t | ≥ 2 . We also provide new bounds that are useful for intermediate values of $$|t|$$ | t | . Combined, our results improve the largest known zero-free region within the critical strip for $$3\cdot 10^{12} \le |t|\le \exp (64.1)$$ 3 · 10 12 ≤ | t | ≤ exp ( 64.1 ) and $$|t| \ge \exp (1000)$$ | t | ≥ exp ( 1000 ) .