论文标题
关于排名一号Gallagherian Prime Geodesic定理的改进
On refinements of rank one Gallagherian prime geodesic theorems
论文作者
论文摘要
在他最近的研究中，作者改善了针对紧凑型，均匀，等级的本地对称空间的主要地球定理中的错误术语。事实证明，获得的估计$ O（x^{2ρ-\fracρ{n}}（\ log x）^{ - 1}）$与最著名的结果相吻合，因为紧凑的Riemann表面，三个流形，三个流形，与Cusps和Cusps相吻合，并与cusps一起使用，其中$ n $代表了空间的空间和$ p $ s的$ p poce，and $ n $是$ p的ump poce。然后将上述绑定降低为$ O（x^{2ρ-ρ\ frac {2 \ cdot（2n）+1} {2n \ cdot（2n）+1}}（\ log x）^{\ frac {\ frac {n-1} x）^{\ frac {n-1} {2n \ cdot（2n）+1}+\ varepsilon}）$在Gallagherian Sense中，带有$ \ varepsilon $ $> $ $ 0 $，以及计数功能$ψ_{2n}（2n}（x）$的关键作用。这项研究的目的是证明后者可以进一步减少。为此，我们为功能的新的显式公式$ψ_{j}（x）$，$ j $ $ \ geq $ $ n $，以及$ψ_{n-1}（x）$的条件公式。应用Gallagher-Koyama技术，我们将$ψ_{0}（x）$的渐近剂和Gallagherian Prime Geodesic定理推断出来。获得的错误项$ O（X^{2ρ-ρ\ frac {2J+1} {2nj+1}}}（\ log x）^{\ frac {n-1} {n-1} {2nj+1} -1} -1} -1} -1}（\ log x） $n-1$ $\leq$ $j$ $<$ $2n$, improve the $O$-term given above, with the optimal unconditional (conditional) size achieved for $j$ $=$ $n$ ($j$ $=$ $n-1$).如果$ j $ = $ = $ n $ $ \ geq $ 4 $，那么我们的新界限与cusps案的流形中最著名的估计相吻合。 If $j$ $=$ $n-1$, the $O$-term fully agrees with the results in the Riemann surface case ($n$ $=$ $2$, $ρ$ $=$ $\frac{1}{2}(n-1)$ $=$ $\frac{1}{2}$), and the three manifolds case ($n$ $=$ $2$, $ρ$ $=$ $ 1 $）。最后，对于$ j $ = $ $ $ n-1 $，$ n $ $ \ geq $ 4 $，我们的结果改善了cusps案中最著名的界限。
In his recent research, the author improved the error term in the prime geodesic theorem for compact, even-dimensional, rank one locally symmetric spaces. It turned out that the obtained estimate $O(x^{2ρ-\fracρ{n}}(\log x)^{-1})$ coincides with the best known results for compact Riemann surfaces, three manifolds, and manifolds with cusps, where $n$ stands for the dimension of the space, and $ρ$ is the half-sum of positive roots. The above bound was then reduced to $O(x^{2ρ-ρ\frac{2\cdot(2n)+1}{2n\cdot(2n)+1}}(\log x)^{\frac{n-1}{2n\cdot(2n)+1}-1}(\log\log x)^{\frac{n-1}{2n\cdot(2n)+1}+\varepsilon})$ in the Gallagherian sense, with $\varepsilon$ $>$ $0$, and the key role played by the counting function $ψ_{2n}(x)$. The purpose of this research is to prove that the latter $O$-term can be further reduced. To do so, we derive new explicit formulas for the functions $ψ_{j}(x)$, $j$ $\geq$ $n$, and conditional formula for $ψ_{n-1}(x)$. Applying the Gallagher-Koyama techniques, we deduce the asymptotics for $ψ_{0}(x)$, and the Gallagherian prime geodesic theorems. The obtained error terms $O(x^{2ρ-ρ\frac{2j+1}{2nj+1}}(\log x)^{\frac{n-1}{2nj+1}-1}(\log\log x)^{\frac{n-1}{2nj+1}+\varepsilon})$, $n-1$ $\leq$ $j$ $<$ $2n$, improve the $O$-term given above, with the optimal unconditional (conditional) size achieved for $j$ $=$ $n$ ($j$ $=$ $n-1$). If $j$ $=$ $n$ $\geq$ $4$, our new bound coincides with the best known estimate in the manifolds with cusps case. If $j$ $=$ $n-1$, the $O$-term fully agrees with the results in the Riemann surface case ($n$ $=$ $2$, $ρ$ $=$ $\frac{1}{2}(n-1)$ $=$ $\frac{1}{2}$), and the three manifolds case ($n$ $=$ $2$, $ρ$ $=$ $1$). Finally, for $j$ $=$ $n-1$, $n$ $\geq$ $4$, our result improves the best known bound in the manifolds with cusps case.
