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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 | Abstract: | Let $S(t) = frac{1}{pi} arg zeta ig(frac{1}{2} + it ig)$ 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 | Services: | Forum | Review | PDF | Favorites |
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