Extreme values for Sn(σ,t) near the critical line

Abstract

Let S(σ, t) =1πargζ(σ+it)be the argument of the Riemann zeta function at the point σ+itof the critical strip. Fo r n ≥1and t >0we defineSn(σ, t)=t∫0Sn−1(σ, τ)dτ+δn,σ,where δn,σis a specific constant depending on σand n. Let 0 ≤β<1be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function Sn(σ, t)on the interval Tβ≤t ≤Tand near to the critical line, when n ≡1mod4. Similar estimates are obtained for |Sn(σ, t)|when n ≡1mod4. This extends the results of Bondarenko and Seip [7]for a region near the critical line. In particular we obtain some omega results for these functions on the critical line.