论文标题
光谱Zeta功能在离散的Tori和Epstein-Riemann猜想上
Spectral zeta function on discrete tori and Epstein-Riemann conjecture
论文作者
论文摘要
我们考虑了一系列离散的Tori的组合拉普拉斯，该序列近似于m维圆环。在特殊情况下，M = 1，弗里德利和卡尔森在临界条中衍生了相应光谱Zeta函数的渐近膨胀，因为近似参数转移到无穷大。在那里，作者还对这种渐近学提出了一个猜想，这等同于Riemann猜想。在本文中，受到弗里德利（Friedli）和卡尔森（Karlsson）的工作的启发，我们证明了类似的渐近膨胀成本也适用于M = 2。类似的论点也适用于更高的维度。如果我们用$ 9 $ - 点星级离散的laplacian替换标准离散的laplacian，则对这种渐近学的猜想给了爱泼斯坦 - 里曼猜想的等效公式。
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 well. A conjecture on this asymptotics gives an equivalent formulation of the Epstein-Riemann conjecture, if we replace the standard discrete Laplacian with the $9$-point star discrete Laplacian.
