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- W4287757617 abstract "Let $S(t) = tfrac{1}{pi} arg zeta big({1/2} + it big)$ be the argument of the Riemann zeta-function at the point $tfrac12 + it$. For $n geq 1$ and $t>0$ define its antiderivatives as begin{equation*} S_n(t) = int_0^t S_{n-1}(tau) hspace{0.08cm} rm dtau + 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 $$ int_{0}^{T}|S_n(t)|^2 hspace{0.06cm} rm dt = O(T), $$ for $ngeq 1$. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of $S(t)$ and $S_1(t)$. This was extended by Fujii for $S_n(t)$, when $ngeq 2$. 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 $ngeq 1$. 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." @default.
- W4287757617 created "2022-07-26" @default.
- W4287757617 creator A5034413436 @default.
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- W4287757617 date "2020-06-15" @default.
- W4287757617 modified "2023-10-16" @default.
- W4287757617 title "The second moment of $S_n(t)$ on the Riemann hypothesis" @default.
- W4287757617 doi "https://doi.org/10.48550/arxiv.2006.08503" @default.
- W4287757617 hasPublicationYear "2020" @default.
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