On Fried's conjecture for compact hyperbolic manifolds

双曲度量附近普通Fried猜想的反例

Counterexamples to the ordinary Fried conjecture near hyperbolic metrics

该研究借助LLM辅助构造三维双曲度量附近普通Fried猜想的反例，证明非循环酉线特征趋于平凡特征时，Ruelle ζ函数零点模与解析挠率平方的比值趋于0。

Fried conjecture in small dimensions

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flo…

有限体积完全双曲型三重体的Iwasawa猜想的一个类似

An analog of the Iwasawa conjecture for a complete hyperbolic threefold of finite volume

对于只有一个尖点的完备双曲型三重有限体积上的秩为一的酉局部系统，我们将比较t=1时Alexander不变量的阶和s=0时Ruelle-Selberg L-函数的阶.我们的结果可以看作是代数数论中岩泽主猜想的一个几何类比。

弗里德曼-拉马努金函数在随机双曲几何中的应用及谱间隙I

Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I

本文研究双曲曲面模空间上长度函数的水平集，引入新的体积函数V_g^T(l)用于计算随机双曲曲面上闭测地线的数量，并证明其渐近展开式，引入弗里德曼-拉马努金函数类，探讨谱间隙问题。

Spectral zeta function on discrete tori and Epstein-Riemann conjecture

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for m=2. Similar argument applies to higher dimensions as we…

Extension of Alon's and Friedman's conjectures to Schottky surfaces

Let $X=Λ\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $δ$ denote the Hausdorff dimension of the limit set of $Λ$. We prove that for any compact subset $\mathcal{K} \subset\{\,s\,:\,\Re(s)>\fracδ{2}\,\}$, if one picks a random degree $n$ cover $X_{n}$ of $X$ uniformly at random, then with probability tending to one as $n\to\infty$, there are no resonances of $X_{n}$ in $\mathcal{K}$ other than those already belonging to $X$ (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequen…