Zeta function for nonpositive integers

Question

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$

Now the Benoulli numbers are defined by $\frac{1}{e^t-1}=\sum_{m=0}^{\infty}B_m\frac{t^{m-1}}{m!}$ where $B_0=1, B_1=1/2, B_{2m+1}=0$

How can I use these things to get an expression for $\zeta(-n), n=0,1,2,3...$ in terms of $B_n$

Answer 1

I assume you mean $$\zeta(z) = \frac{\Gamma(1-z)}{2 \pi i} \int_{C} \frac{t^{z-1} }{e^{-t}-1} \, \mathrm dt = -\frac{\Gamma(1-z)}{2 \pi i} \int_{C} \frac{t^{z-1}e^{t} }{e^{t}-1} \, \mathrm dt,$$

where $C$ is a contour that starts at $- \infty$ below the branch cut on the negative real axis, goes around the origin (without enclosing any of the points $z= \pm 2 \pi i, \pm 4 \pi i, \ldots$), and then goes to back to $-\infty$ above the branch cut.

The above integral representation of the Riemann zeta function is valid for all complex values of $z$ except positive integers.

If we let $z= - n$, where $n \in \mathbb{N}_{\ge 0}$, we get

$$ \zeta(-n) = -\frac{\Gamma(n+1)}{2 \pi i} \int_{C} \frac{t^{-n-1}e^{t}}{e^{t}-1} \, \mathrm dt.$$

But since $z$ is now an integer, the integral above and below the branch cut cancel each other, and all we're left with is the circle around the origin.

Therefore,

$$ \begin{align} \zeta(-n) &= -\frac{\Gamma(n+ 1)}{2 \pi i } \, 2 \pi i \, \operatorname{Res}_{t=0} \left(\frac{t^{-n-1} e^{t}}{e^{t}-1} \right) \\ &= - n! \, \operatorname{Res}_{t=0} \left( t^{-n-1} \sum_{m=0}^{\infty} \frac{B_{m}(1)}{m!} t^{m-1} \right) \tag{1}\\ & = - n! \, \frac{B_{n+1}(1)}{(n+1)!} \\ &= - \frac{B_{n+1}(1)}{n+1} \\&= (-1)^{n} \frac{B_{n+1}}{n+1}. \end{align}$$

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$(1)$ https://en.wikipedia.org/wiki/Bernoulli_polynomials#Generating_functions

Answer 2

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle #1 \right\rangle} \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace} \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left( #1 \right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $$ \mbox{Use}\quad\zeta\pars{-n}= -2^{-n}\pi^{-n - 1}\Gamma\pars{1 + n}\zeta\pars{1 + n}\sin\pars{n\pi \over 2} $$

Answer 3

We have for $s$ with $\Re{e}(s) > 1$ :

$\Gamma(s)\zeta(s)=\int_0^\infty\frac{t^{s-1}}{{\rm e}^t-1}\mathrm dt=\int_0^1\frac{t^{s-1}}{\mathrm e^t-1}\mathrm dt+\int_1^\infty\frac{t^{s-1}}{\mathrm e^t-1} dt$.

The second integral is holomorphe in $s$. We take the Taylor series in the first integral.

We have for all $t$ with $|t| < 2\pi$

$\frac t{{\rm e}^t-1}=\sum_{n=0}^\infty\frac{B_n t^n}{n!}$,

where the $B_n$ are the Bernoulli's numbers. Integrating term by term

$\zeta(s)=\frac1{(s-1)\Gamma(s)}+\frac1{\Gamma(s)}\sum_{n=1}^\infty\frac{B_n}{n!(n+s-1)}+\frac1{\Gamma(s)}\int_1^\infty\frac{t^{s-1}}{\mathrm e^t-1} \mathrm dt$.

The series converge and is holomorphic for all s but $s=-n$, $(n \in \mathbb{N})$ because the convergence radius of the series is not modified by division by $n + s – 1$.

When $s \to k$, as $Γ(s)$ has a simple pole in $s=–k$ , $ζ(s)$ is the sum of one term which tend to $0$ and $\frac1{\Gamma(s)}~\frac{B_{k+1}}{(k+1)!~(s+k)}~\underset{\overset{s\to-k}{}}{\sim}~\frac{s+k}{\text{Res}(\Gamma,-k)}~\frac{B_{k+1}}{(k+1)!~(s+k)}=(-1)^k~k!~\frac{B_{k+1}}{(k+1)!}.$

We have the Euler formula : $\zeta(-k)=(-1)^k\frac{B_{k+1}}{k+1}.$