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29 March 2024
 
  » arxiv » 2006.8503

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The second moment of $S_n(t)$ on the Riemann hypothesis
Andrés Chirre ; Oscar E. Quesada-Herrera ;
Date 15 Jun 2020
AbstractLet $S(t) = frac{1}{pi} arg zeta ig(frac{1}{2} + it ig)$ be the argument of the Riemann zeta-function at the point $ frac12 + it$. For $n geq 1$ and $t>0$ define its antiderivatives as egin{equation*} S_n(t) = int_0^t S_{n-1}( au) hspace{0.08cm} m d au + delta_n , end{equation*} where $delta_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that $$ int_{0}^{T}|S_n(t)|^2 hspace{0.06cm} m dt = O(T), $$ for $ngeq 1$. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of $S(t)$ and $S_1(t)$. 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 $ngeq 1$. Our result conditionally refines Selberg’s formula for $S_1(t)$ and makes explicit the estimates given by Littlewood. This extends previous work by Goldston in $1987$, where the case $n=0$ was considered.
Source arXiv, 2006.8503
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