The second moment of S_n(t) on the Riemann hypothesis

2020/06/15 by Chirre, Andrés, Quesada-Herrera, Emily
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.08503

Abstract

Let S(t) = \tfrac1π arg ζ(1/2 + it ) be the argument of the Riemann zeta-function at the point \tfrac12 + it. For n ≥ 1 and t>0 define its antiderivatives as S_n(t) = ∫₀^t S_n-1(τ) \hspace0.08cm \rm dτ+ δ_n, where δ_n is a specific constant depending on n and S₀(t) := S(t). In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that ∫₀^T|S_n(t)|² \hspace0.06cm \rm dt = O(T), for n≥ 1. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of S(t) and S₁(t). This was extended by Fujii for S_n(t), when n≥ 2. Assuming the Riemann Hypothesis, we give the explicit asymptotic formula for the second moment of S_n(t) up to the second-order term, for n≥ 1. Our result conditionally refines Selberg's and Fujii's formulas and extends previous work by Goldston in 1987, where the case n=0 was considered.

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