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

Abstract

Let S(σ,t)=1πarg⁡ζ(σ+it) be the argument of the Riemann zeta function at the point σ+it of the critical strip. For n≥1 and t>0 we defineSn(σ,t)=∫0tSn−1(σ,τ)dτ+δn,σ, where δn,σ is a specific constant depending on σ and n. Let 0≤β<1 be 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≤T and 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.