循环图的Zeta函数
The zeta function for circular graphs
摘要：
研究了整函数zeta（n，s），它是l的-s次方和，其中l遍历n阶循环图C（n）的Laplacian的正特征值.我们证明了zeta（n，s）的根对n无穷大收敛于直线Re（s）=1/2，在这个意义上，对于这条直线的补线中的每个紧子集K，并且n足够大，zeta函数zeta（n，s）没有根在K中。为了证明这一点，我们来看看狄拉克zeta函数，它使用了循环图的狄拉克算子D=d+d^* 的正特征值，即拉普拉斯算子的平方根。我们扩展了牛顿-科茨-罗尔型分析的黎曼和，并使用一个衍生工具，具有相似之处的Schwarzian衍生。由于圆图的zeta函数zeta（n，s）是整函数，所以这个结果并没有说明经典的黎曼ζ函数zeta（s）的根，它也是圆的Dirac zeta函数。只有当Re（s）>1时，zeta（n，s）的值才适当地收敛到zeta（s）。我们还给出了离散巴塞尔问题的一个新的解，它给出了类似zeta_n（2）=（n^2-1）/12或zeta_n（4）=（n^2-1）（n^2+11）/45的表达式，这使得经典巴塞尔问题zeta（2）= pi^2/6或zeta（4）=pi^4/90在连续极限下的值可以重新导出.
英文摘要：
We study the entire function zeta(n,s) which is the sum of l to the power -s, where l runs over the positive eigenvalues of the Laplacian of the circular graph C(n) with n vertices. We prove that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(n,s) is in K. To prove this, we look at the Dirac zeta function, which uses the positive eigenvalues of the Dirac operator D=d+d^* of the circular graph, the square root of the Laplacian. We extend a Newton-Coates-Rolle type analysis for Riemann sums and use a derivative which has similarities with the Schwarzian derivative. As the zeta functions zeta(n,s) of the circular graphs are entire functions, the result does not say anything about the roots of the classical Riemann zeta function zeta(s), which is also the Dirac zeta function for the circle. Only for Re(s)>1, the values of zeta(n,s) converge suitably scaled to zeta(s). We also give a new solution to the discrete Basel problem which is giving expressions like zeta_n(2) = (n^2-1)/12 or zeta_n(4) = (n^2-1)(n^2+11)/45 which allows to re-derive the values of the classical Basel problem zeta(2) = pi^2/6 or zeta(4)=pi^4/90 in the continuum limit.
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