arXiv 1702.040992017-02-14math.NTmath.CA

黎曼假设上的$S_n(t)$

Bounding $S_n(t)$ on the Riemann hypothesis

Emanuel Carneiro, Andrés Chirre

备注：: 23 pages. To appear in Math. Proc. Cambridge Philos. Soc
发表信息：: Mathematical Proceedings of the Cambridge Philosophical Society, vol.164 (2018), 259-283
DOI：: 10.1017/S0305004116000992

摘要：

设$S(t) = \tfrac{1}π \arg ζ(\frac12 + it)$为黎曼ζ函数在点$\tfrac12 + it$处的辐角。对于$n \geq 1$和$t>0$，定义其迭代\begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \,{\rm d}τ\, + δ_n\,, \end{equation*}，其中$δ_n$是取决于$n$和$S_0(t) := S(t)$的特定常数。1924年，J.E. Littlewood证明，根据黎曼假设（RH），$S_n(t) = O(\log t/ (\log \log t)^{n+1})$。迄今为止，这一估计的数量级从未得到改善。$S(t)$和$S_1(t)$的最佳边界目前是由于Escheriro，Chandee和Milinovich。在本文中，我们建立，RH下，这个估计\begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*}的一个明确的形式，所有$n\geq 2$，常数$C_n^{\pm}$指数衰减快$n \to \infty$。这改善了（所有$n \geq 2$）的结果若狭，谁以前获得了这样的界限与常数趋于一个固定的值时，$n \to \infty$。我们的方法使用特殊的极值函数的指数型来自高斯从属框架的Escheriro，Littmann和Vaaler的$n$是奇数的情况下，$n$是偶数的情况下，优化的插值参数。在最后一节中，我们将这些结果推广到一般的$L$ -函数类。

英文摘要：

Let $S(t) = \tfrac{1}π \arg ζ(\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \,{\rm d}τ\, + δ_n\,, \end{equation*} where $δ_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that $S_n(t) = O(\log t/ (\log \log t)^{n+1})$. The order of magnitude of this estimate was never improved up to this date. The best bounds for $S(t)$ and $S_1(t)$ are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate \begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*} for all $n\geq 2$, with the constants $C_n^{\pm}$ decaying exponentially fast as $n \to \infty$. This improves (for all $n \geq 2$) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when $n \to \infty$. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when $n$ is odd, and an optimized interpolation argument for the cases when $n$ is even. In the final section we extend these results to a general class of $L$-functions.

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