THE LARGE VALUES OF THE RIEMANN ZETA-FUNCTION

Let Absolute value of theta < pi/2 and sigma is-an-element-of [1/2, 1] . By refining Selberg's method, we study the large values of Re {e(-it heta) log zeta(sigma + it)} as t --> infinity. For sigma close to 1/2 we obtain OMEGA+ estimates that are as good as those obtained previous ly on the Riemann Hypothesis. In particular, we show that (sup(T < t l ess-than-or-equal-to 2T) log \zeta(1/2 + it)\)(sup(T < t less-than-or- equal-to 2T) +/-S(t)) much greater than T/log log T and S1(t) = OMEGA((log t)1/2(log log t)-3/2). Our results supplement those of Montgomer y which are good when sigma > 1/2 is fixed.