Green’s Functions and Euler’s Formula for $\zeta (2n)$

Abstract

In this note, we calculate the Green’s function for the linear operator $(- \Delta _D)^n,$ where $- \Delta _D$ is the one-dimensional Dirichlet Laplacian in $L^2((0,1); dx)$ defined by

$$\begin{aligned} - \Delta _D f=-f^{\prime \prime } \end{aligned}$$

with (Dirichlet) boundary conditions $f(0)=f(1)=0.$ As a consequence of this computation, we obtain Euler’s formula

$$\begin{aligned} \zeta (2n) = \sum _{k \in {\mathbb {N}}} k^{-2n} = \dfrac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}, \quad n\in {\mathbb {N}}, \end{aligned}$$

where $\zeta (\cdot )$ denotes the Riemann zeta function and $B_{n}$ is the nth Bernoulli number. This generalizes the example given by Grieser [29] for $n=1.$ In addition, we derive its z-dependent generalization for $z \in {\mathbb {C}}\backslash \big \{(k \pi )^{2n}\big \}_{k \in {\mathbb {N}}}$,

$$\begin{aligned} \sum _{k \in {\mathbb {N}}} \big [(k\pi )^{2n} - z\big ]^{-1} = \frac{1}{2n z}\bigg [n - \sum _{j=0}^{n-1} \omega _j^{1/2} z^{1/(2n)} \cot \big (\omega _j^{1/2} z^{1/(2n)}\big )\bigg ], \quad n\in {\mathbb {N}}, \end{aligned}$$

where $\omega _j = e^{2 \pi i j/n}$, $0 \le j \le n-1$, represent the nth roots of unity. In this context we also derive the Green’s function of $\big ((- \Delta _D)^n - z I\big )^{-1}$, $n \in {\mathbb {N}}$.

Dedicated to the memory of Erik Balslev (9-27-1935–1-11-2013). K. K. was supported by the Baylor University Summer Sabbatical and Research Leave Program.