论文标题
正则化ZETA功能和扩散过程的功能方程
Functional equations for regularized zeta-functions and diffusion processes
论文作者
论文摘要
我们讨论了Riemann Zeta功能的整体表示中的修改，该函数导致了Riemann功能方程的概括，该方程可保留对称性$ s \ to（1-S）$（1-S）$。通过使用表现出对称性$ x \ mapsto 1/x $的截止值修改zeta功能的一个积分表示，我们获得了涉及第二种贝塞尔函数的广义功能方程。接下来，在另一个表现出相同对称性的截止值的情况下，我们获得了仅涉及第二种贝塞尔函数的功能方程的概括。讨论了欧几里得和双曲线空间的一个正则Zeta功能与热核的拉普拉斯变换之间的联系。
We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry $x\mapsto 1/x$, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.
