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

2018/07/31 by Chirre, Andrés
#11M06 #11M26 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.11642

Abstract

Let S(σ,t)=\frac1πargζ(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 define S_n(σ,t) = ∫₀^t S_n-1(σ,τ) dτ + δ_n,σ , where δ_n,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 S_n(σ,t) on the interval T^β≤ t ≤ T and near to the critical line, when n≡ 1\mod 4. Similar estimates are obtained for |S_n(σ,t)| when n\not≡ 1\mod 4. This extends a recently results of Bondarenko and Seip for a region near the critical line. In particular we obtain some omega results for these functions on the critical line.

Related

• Large values of the argument of the Riemann zeta-function and its iterates
• On Large Values of |ζ(σ+\rm it)|
• Extreme values of the Riemann zeta function at its critical points in the critical strip
• Explicit bounds on ζ(s) in the critical strip and a zero-free region
• Extreme values of the Dirichlet L-functions at the critical points of the Riemann zeta function