The large values of the Riemann Zeta‐function

Let |θ| < π/2 and σ ∈ [ 1 2 ,   1 ] . By refining Selberg's method, we study the large values of R e { e − i θ   log   ζ ( σ + i t ) }as t → ∞ For σ close to ½ we obtain Ω + estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that(sup T < t ⩽ 2 T   log   | ζ ( 1 2 + i t ) | ) (sup T < t ⩽ 2 T ± S ( t ) ) ≫ log   T / log   log   TandS 1 ( t ) = Ω + (( log   t )1 / 2( log   log   t )− 3 / 2) .Our results supplement those of Montgomery which are good when σ > ½ is fixed.