论文标题
Riemann假设上的$ s_n（t）$的第二时
The second moment of $S_n(t)$ on the Riemann hypothesis
论文作者
论文摘要
令$ s（t）= \ tfrac {1}π\ arg amζ\ big（{1/2} + it \ big）$是$ \ tfrac12 + IT $的riemann zeta-函数的参数。对于$ n \ geq 1 $和$ t> 0 $将其抗动力定义为\ begin {qore*} s_n（t）= \ int_0^t s_ {n-1}（n-1}（τ）\ hspace {0.08cm} {0.08cm} $ s_0（t）：= s（t）$。 1925年，J。E。Littlewood在Riemann假设下证明了$$ \ int_ {0}^{t} | s_n（t）|^2 \ hspace {0.06cm} \ rm dt = o（t），$ n \ geq 1 $。 1946年，塞尔伯格无条件地建立了$ s（t）$和$ s_1（t）$的第二瞬间的显式渐近公式。 Fujii以$ s_n（t）$的价格扩展了这一点，当$ n \ geq 2 $时。假设Riemann假设，我们给出了$ s_n（t）$的第二阶段的显式渐近公式，直到二阶期限为$ n \ geq 1 $。我们的结果有条件地完善了Selberg和Fujii的公式，并扩展了1987年Goldston的先前工作，其中考虑了$ n = 0 $的情况。
Let $S(t) = \tfrac{1}π \arg ζ\big({1/2} + it \big)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its antiderivatives as \begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \hspace{0.08cm} \rm dτ+ δ_n, \end{equation*} where $δ_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that $$ \int_{0}^{T}|S_n(t)|^2 \hspace{0.06cm} \rm dt = O(T), $$ for $n\geq 1$. 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 $n\geq 2$. 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 $n\geq 1$. 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.
