A question that has been puzzling me for quite some time now:
Why is the value of the Riemann Zeta function equal to $0$ for every even negative number?
I assume that even negative refers to the real part of the number, while its imaginary part is $0$.
So consider $-2$ for example:
$f(-2) = \sum_{n=1}^{\infty}\frac{1}{n^{-2}} = \frac{1}{1^{-2}}+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+\dots = 1^2+2^2+3^2+\dots = \infty$
What am I missing here?
For $\mathrm{Re}(s)\gt1$, $$ \zeta(s)=\sum_{n=1}^\infty\frac1{n^s}\tag{1} $$ For $\mathrm{Re}(s)\le1$, we need to use Analytic Continuation and another formula for $\zeta(s)$ since $(1)$ does not converge for $\mathrm{Re}(s)\le1$.
One formula that can be used is the Functional Equation for $\zeta$ which says $$ \zeta(s)\frac{\Gamma(s/2)}{\pi^{s/2}}=\zeta(1-s)\frac{\Gamma((1-s)/2)}{\pi^{(1-s)/2}}\tag{2} $$ The symmetric form in $(2)$ is given as $(14)$ on MathWorld.
Since the recurrence for $\Gamma$ says that for $n\in\mathbb{Z}$ and $n\le0$, $\frac1{\Gamma(n)}=0$, we get that $\zeta(2n)=0$ for $n\in\mathbb{Z}$ and $n\lt0$.
Analytic Continuation of $\boldsymbol{\zeta}$ Using $\boldsymbol{\eta}$ and Integration by Parts
Define $\eta$, the alternating $\zeta$ function, as $$ \begin{align} \eta(s) &=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}\\ &=\sum_{n=1}^\infty\frac1{n^s}-2\sum_{n=1}^\infty\frac1{(2n)^s}\\[6pt] &=\zeta(s)\left(1-2^{1-s}\right)\tag{3} \end{align} $$ Formula $(3)$ follows since the terms of an alternating series are the terms of the non-alternating series minus twice the even terms.
Formula $(3)$ increases the domain to $\operatorname{Re}(s)\gt0$. $\eta$ also comes in handy to define $$ \eta(s)\,\Gamma(s)=\int_0^\infty\frac{t^{s-1}}{e^t+1}\,\mathrm{d}t\tag{4} $$ which also converges for $\operatorname{Re}(s)\gt0$. However, we can integrate $(4)$ by parts $k$ times to get $$ \bbox[5px,border:2px solid #C0A000]{\eta(s)\,\Gamma(s)=\frac{(-1)^k}{s(s+1)\cdots(s+k-1)}\int_0^\infty t^{s+k-1}\frac{\mathrm{d}^k}{\mathrm{d}t^k}\frac1{e^t+1}\,\mathrm{d}t}\tag{5} $$ $(5)$ agrees with $(4)$ for $\operatorname{Re}(s)\gt0$ and converges for $\operatorname{Re}(s)\gt-k$. Thus, $(5)$ gives an analytic continuation of $\zeta(s)$ for $\operatorname{Re}(s)\gt-k$.
Using this method, $\zeta(0)=-\frac12$ is computed in this answer and $\zeta(-1)=-\frac1{12}$ is computed in this answer.
Proving $\boldsymbol{\zeta(-2n)=0}$
Note that $(5)$ can be rewritten as $$ \eta(s)\,\Gamma(s+k)=(-1)^k\int_0^\infty t^{s+k-1}\frac{\mathrm{d}^k}{\mathrm{d}t^k}\left(\frac12-\frac12\frac{e^{t/2}-e^{-t/2}}{e^{t/2}+e^{-t/2}}\right)\,\mathrm{d}t\tag{6} $$ Setting $s=1-k$ in $(6)$ gives $$ \begin{align} \eta(1-k) &=(-1)^k\int_0^\infty\frac{\mathrm{d}^k}{\mathrm{d}t^k}\left(\frac12-\frac12\frac{e^{t/2}-e^{-t/2}}{e^{t/2}+e^{-t/2}}\right)\,\mathrm{d}t\\[6pt] &=(-1)^{k-1}\frac{\mathrm{d}^{k-1}}{\mathrm{d}t^{k-1}}\left.\left(\frac12-\frac12\frac{e^{t/2}-e^{-t/2}}{e^{t/2}+e^{-t/2}}\right)\right|_{t=0}\tag{7} \end{align} $$ Set $k=2n+1$ and we get $$ \eta(-2n) =\frac{\mathrm{d}^{2n}}{\mathrm{d}t^{2n}}\left.\left(\frac12-\frac12\frac{e^{t/2}-e^{-t/2}}{e^{t/2}+e^{-t/2}}\right)\right|_{t=0}\tag{8} $$ For $n\ge1$, the right side of $(8)$ is an odd function, so evaluating at $t=0$, and applying $(3)$, yields $$ \bbox[5px,border:2px solid #C0A000]{\zeta(-2n)=0}\tag{9} $$
Analytic Continuation of $\boldsymbol{\zeta}$ Using the Euler-Maclaurin Sum Formula
As described in this answer, this answer, and this answer, for $\operatorname{Re}(s)\gt-2m-1$, $\zeta(s)$ can be represented as $$ \hspace{-12pt}\bbox[5px,border:2px solid #C0A000]{\zeta(s)=\lim_{n\to\infty}\left[\sum_{k=1}^n\frac{1}{k^s}-\left(\frac{1}{1-s}n^{1-s}+\frac12n^{-s}-\sum_{k=1}^m\frac{B_{2k}}{2k}\binom{s+2k-2}{2k-1}n^{-s-2k+1}\right)\right]}\tag{10} $$ where the formula in the parentheses is obtained from the Euler-Maclaurin Sum Formula.
Proving $\boldsymbol{\zeta(-2m)=0}$
If we plug $s=-2m$ into $(10)$, for $m\ge1$, the formula in parentheses exactly matches the sum outside the parentheses by Faulhaber's Formula. In fact, the Euler-Maclaurin Sum Formula is one way to prove Faulhaber's Formula. This means that $$ \bbox[5px,border:2px solid #C0A000]{\zeta(-2m)=0}\tag{11} $$ We can also plug $s=-2m+1$, for $m\ge1$, into $(10)$ and note that Faulhaber's Formula does not include the constant term. Thus, we get $$ \zeta(-2m+1)=-\frac{B_{2m}}{2m}\tag{12} $$
Functional Equation for $\boldsymbol{\zeta}$
The Fourier Transform of $e^{-\pi x^2t}$ is $$ \begin{align} \int_{-\infty}^\infty e^{-\pi x^2t}e^{-2\pi ix\xi}\,\mathrm{d}x &=\int_{-\infty}^\infty e^{-\pi(x-i\xi/t)^2t}e^{-\pi\xi^2/t}\,\mathrm{d}x\\ &=\frac1{\sqrt{t}}e^{-\pi\xi^2/t}\tag{13} \end{align} $$ Applying the Poisson Summation Formula to $(13)$ says that $$ 1+2\sum_{n=1}^\infty e^{-\pi n^2t} =\frac1{\sqrt{t}}+\frac2{\sqrt{t}}\sum_{n=1}^\infty e^{-\pi n^2/t}\tag{14} $$ Note that for $s\gt1$, $$ \begin{align} &\zeta(s)\frac{\Gamma(s/2)}{\pi^{s/2}}\\ &=\sum_{n=1}^\infty\frac1{n^s}\int_0^\infty e^{-\pi t}t^{\frac{\large s}2}\,\frac{\mathrm{d}t}t\\ &=\int_0^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)t^{\frac{\large s}2}\,\frac{\mathrm{d}t}t\\ &=\int_0^1\left(\frac1{2\sqrt{t}}-\frac12+\frac1{\sqrt{t}}\sum_{n=1}^\infty e^{-\pi n^2/t}\right)t^{\frac{\large s}2}\,\frac{\mathrm{d}t}t +\int_1^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)t^{\frac{\large s}2}\,\frac{\mathrm{d}t}t\\ &=-\frac1{1-s}-\frac1s+\int_1^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)t^{\frac{1-\large s}2}\,\frac{\mathrm{d}t}t +\int_1^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)t^{\frac{\large s}2}\,\frac{\mathrm{d}t}t\\ &=-\frac1{1-s}-\frac1s+\int_1^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)\left(t^{\frac{1-\large s}2}+t^{\frac{\large s}2}\right)\,\frac{\mathrm{d}t}t\tag{15} \end{align} $$ The following integral is increasing in $\alpha$. For $\alpha\ge0$, we have $$ \begin{align} \int_1^\infty\left(\sum_{n=1}^\infty e^{-\pi n^2t}\right)t^\alpha\,\mathrm{d}t &=\sum_{n=1}^\infty\frac1{\pi^{\alpha+1}n^{2\alpha+2}}\int_{\pi n^2}^\infty e^{-t}\,t^\alpha\,\mathrm{d}t\\ &\le\sum_{n=1}^\infty\frac1{\pi^{\alpha+1}n^{2\alpha+2}}\left(\int_{\pi n^2}^\infty e^{-t}\,\mathrm{d}t\right)^{1/2}\left(\int_{\pi n^2}^\infty e^{-t}t^{2\alpha}\,\mathrm{d}t\right)^{1/2}\\ &\le\frac{\Gamma(2\alpha+1)^{1/2}}{\pi^{\alpha+1}}\sum_{n=1}^\infty\frac{e^{-\pi n^2/2}}{n^{2+2\alpha}}\\ &\le\frac{\Gamma(2\alpha+1)^{1/2}}{\pi^{\alpha+1}}\sum_{n=1}^\infty\frac2{\pi n^{4+2\alpha}}\\ &=\frac{2\,\Gamma(2\alpha+1)^{1/2}}{\pi^{\alpha+2}}\zeta(4+2\alpha)\tag{16} \end{align} $$ Thus, the last integral in $(15)$ defines an entire function. Therefore $(15)$ defines $\zeta(s)\frac{\Gamma(s/2)}{\pi^{s/2}}$ for all $s\in\mathbb{C}$. Since $(15)$ is invariant under $s\leftrightarrow1-s$, we have $$ \bbox[5px,border:2px solid #C0A000]{\zeta(s)\frac{\Gamma(s/2)}{\pi^{s/2}} =\zeta(1-s)\frac{\Gamma((1-s)/2)}{\pi^{(1-s)/2}}}\tag{17} $$
The zeta function is defined as a meromorphic function given by
$$\zeta(s):=1^{-s}+2^{-s}+3^{-s}+\dots=\sum_{n=1}^\infty\frac1{n^s}\ \forall\ \Re(s)>1$$
Though it is common misconception to apply this definition whenever $\Re(s)\le1$, since clearly,
$$1+2+3+\dots$$
doesn't make much sense, normally. To define things like $\zeta(0)$, we use something called analytic continuation (very important part), which allows us to make sense of things. For example,
$$\zeta(s)=\frac1{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^s}\ \forall\ \Re(s)>0\tag!$$
which is not only true for $\Re(s)>1$ but also defined for $\Re(s)>0$. Here, we find that
$$\zeta(0)=\lim_{s\to0^+}\frac1{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^s}=-\frac12$$
We also have the nice reflection formula
$$\zeta(-s)=2^{-s}\pi^{-(s+1)}\sin\left(\frac{-s\pi}2\right)\Gamma(s+1)\zeta(s+1)$$
which can be used to show that
$$\zeta(-2k)=0\ \forall\ k\in\{1,2,3,\dots\}$$
As a short little proof of $(!)$ above, see that if we have
$$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^s}\tag{converges for $\Re(s)>0$}$$
then, by subtracting out the odd terms and adding in the even terms, we are left with double the even terms:
$$\zeta(s)-\eta(s)=\sum_{n=1}^\infty\frac2{(2n)^s}=2^{1-s}\sum_{n=1}^\infty\frac1{n^s}=2^{1-s}\zeta(s)$$
$$\zeta(s)-\eta(s)=2^{1-s}\zeta(s)$$
$$(1-2^{1-s})\zeta(s)=\eta(s)$$
$$\zeta(s)=\frac1{1-2^{1-s}}\eta(s)$$
And that is how one can expand the domain of the zeta function.
Remark: Deriving other representations of the zeta function are usually much more hairy than above and often times require more areas of mathematics than you might first think. However, if you are so inclined, taking the Taylor expansion of $f(s)=s\zeta(s+1)$ should provide a series form of the zeta function converging almost everywhere.
The Zeta function is defined as $\zeta(s)=\sum_{n\ge1}n^{-s}$ only for $s\in\mathbb{C}$ with $\Re(s)>1$!
The function on the whole complex plane (except a few poles) is the analytic continuation of that function.
On the Wikipedia page, you can find the formula: $$\zeta(s)=\frac{2^{s-1}}{s-1}-2^s\int_0^\infty\frac{\sin(s\arctan t)}{(1+t^2)^{\frac{s}{2}}(e^{\pi t}+1)}dt$$ for $s\neq 1$. Maybe working on this integral for $s$ a negative integer will give you the result.
There is a way to prove that $\zeta(-2K) = 0$ :
By definition of the Bernouilli numbers $\frac{z}{e^z-1}= \sum_{k=0}^\infty \frac{B_k}{k!}z^k$ is analytic on $|z| < 2\pi$
Note that $ \frac{z}{e^z-1}-\frac{z}{2} = \frac{z}{2}\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}$ is an even function, therefore $\frac{z}{e^z-1}-1-\frac{z}{2}=\sum_{k=2}^\infty \frac{B_k}{k!}z^k$ is an even function, and $B_{2k+1}=0$ for $k\ge 1$.
For $Re(s) > 0$, let $\Gamma(s) = \int_0^\infty x^{s-1} e^{-x}dx$. It converges absolutely so it is analytic.
Integrating by parts $\Gamma(s+1) = s \Gamma(s)$ providing the analytic continuation to $Re(s) \le 0$ : $\Gamma(s) = \frac{\Gamma(s+k)}{\prod_{m=0}^{k-1} s+m}$.
Thus $\Gamma(s)$ is analytic on $\mathbb{C} \setminus -\mathbb{N}$ with poles at the negative integers where $\Gamma(s) \sim \frac{(-1)^k}{k!}\frac{1}{s+k}$
With the change of variable $x = ny$ you have $\Gamma(s) n^{-s} = \int_0^\infty x^{s-1} e^{-nx}dx$ so that for $Re(s) > 1$ where everything converges absolutely $$\Gamma(s) \zeta(s) = \sum_{n=1}^\infty \int_0^\infty x^{s-1} e^{-nx}dx= \int_0^\infty x^{s-1}\sum_{n=1}^\infty e^{-nx}dx=\int_0^\infty x^{s-2}\frac{x}{e^x-1}dx$$
Note that $\frac{1}{s+k-1} = \int_0^1 x^{s-2+k}dx = \int_0^\infty x^{s-2} x^{k}1_{x < 1}dx$ so that $$\Gamma(s) \zeta(s)- \sum_{k=0}^{K}\frac{B_k}{k!}\frac{1}{s+k-1} =\int_0^\infty x^{s-2}\left(\frac{x}{e^x-1}-\sum_{k=0}^K \frac{B_k}{k!}x^k1_{x < 1}\right)dx \tag{1}$$ Now as $x \to 0$ :$\frac{x}{e^x-1}-\sum_{k=0}^K \frac{B_k}{k!}x^k\sim \frac{B_{K+1}}{(K+1)!}x^{K+1}$ and hence $(1)$ converges and is bounded for $ Re(s) > -K$, i.e. as $s \to -k$ : $$\frac{(-1)^k}{k!}\frac{1}{s+k} \zeta(s) \sim\Gamma(s) \zeta(s)\sim \frac{B_{k+1}}{(k+1)!}\frac{1}{s+k} $$ whence $$\boxed{\zeta(-k) = (-1)^k\frac{B_{k+1}}{k+1} \implies \zeta(-2k) = 0, k \in \mathbb{N}^*}$$
As others have pointed out, that's not quite the definition of the zeta function. The zeta function is in fact the unique meromorphic function that's equal to that wherever that exists. (To prove uniqueness, you can use Taylor series and the theorem that such a function is equal on any disc where it exists to the Taylor series at the center.)
Regarding the given value, I'll give Riemann's original proof:
$$\Gamma(x) = \int_0^\infty\frac{t^{x-1}}{e^t}dt = \int_0^\infty\frac{(nt)^{x-1}}{e^{nt}}ndt = n^x\int_0^\infty \frac{t^{x-1}}{e^{nt}}dt$$
Therefore
$$\Gamma(x)\zeta(x) = \sum_{n=1}^\infty\frac{\Gamma(x)}{n^x} = \sum_{n=1}^\infty\int_0^\infty \frac{t^{x-1}}{e^{nt}}dt = \int_0^\infty t^{x-1}\sum_{n=1}^\infty\frac{1}{(e^{t})^n}dt = \int_0^\infty\frac{t^{x-1}}{e^{t} - 1}dt$$
Now for the slightly tricky part: this still isn't defined for $\Re[x] < 1$. To get around this, what he did was use the identity $2i\sin(\pi x) = e^{i\pi x}-e^{-i\pi x}$ to get
$$2i\sin(\pi x)\Gamma(x)\zeta(x) = (e^{i\pi x}-e^{-i\pi x})\int_0^\infty\frac{t^{x-1}}{e^{t} - 1}dt = \int_0^\infty\frac{(e^{-i\pi}t)^{x-1}}{e^{t} - 1}dt - \int_0^\infty\frac{(e^{i\pi}t)^{x-1}}{e^{t} - 1}dt$$
(Note I've left $e^{\pm i\pi}$ unsimplified, since you'll get different values for each when x isn't an integer.) What Riemann did was he treated this as a contour integral, replacing the hairpin turn at the origin with a tight loop - which doesn't change the value since the integrand is analytic everywhere in the origin's neighborhood. So what you ultimately have is
$$\zeta(x) = \lim_{s\to x}\frac{\int_C\frac{(-t)^{s-1}}{e^t - 1}dt}{2i\sin(\pi s)\Gamma(s)}$$
Where the contour treats -1 as $e^{i\pi}$ on the way in, $e^{-i\pi}$ on the way out, and goes around the origin in a tight loop in a clockwise (i.e., negative) direction. Fortunately, in the case of integers positive or negative, there's no branch cut, so the way in and the way out cancel, leaving only a tight loop that can be computed with the residue theorem. So with that in mind,
$$\int_C\frac{(-t)^{s-1}}{e^t - 1}dt = -\int_C (-t)^{s-2}\frac{t}{e^t - 1}dt$$
It turns out that the coefficients of the Maclaurin expansion of $\frac{x}{e^x-1}$ are just the Bernoulli numbers divided by the factorials, and every other Bernoulli number starting with the fourth is zero - meaning that the coefficient of every odd exponent of the Maclaurin series but $x^1$ is zero, so where s is a negative even integer, the residue is zero, so the integral is zero.
So there's still one hurdle - the gamma function has poles at nonpositive integers, and $\sin(\pi x)$ is equal to zero. That's easy enough to work out, though, using the property $s\Gamma(s) = \Gamma(s+1)$ (which if you didn't know already, integration by parts will show you is implied by the definition above), and the fact that x is a negative even integer,
$$\lim_{s\to x}\frac{\frac{1}{\Gamma(s)}}{\sin(\pi s)} = \lim_{s\to x}\left(\frac{\prod_{n=0}^{-x}(s-x-n)}{(s-x)\Gamma(s-x+1)}\cdot\frac{(s-x)}{\sin(\pi s)}\right) = \frac{(-x)!}{\pi}$$
What this is isn't important, only that now you have zero in the numerator and a nonzero denominator, so you have zero.
Here is some nice info: https://en.wikipedia.org/wiki/Riemann_zeta_function#The_functional_equation
To answer your question, there is a functional equation that the zeta function satisfies: $$\zeta(s)=2^s\pi^{s-1}\sin\Big( \frac{\pi s}{2} \Big) \Gamma(1-s)\zeta(1-s),$$ where $\Gamma$ is the Gamma function. For negative even integers $k$ (-2, -4 et cetera), the sine factor is zero, which makes $\zeta(k)=0$.
Other answers have addressed in detail the specifics of the zeta function. This is just to add a simple note which, I think, goes to the heart of the OP's question as to what he or she is missing, namely how can it make sense to talk about the behavior of a function defined by an infinite series at a value where the series diverges.
The simplest way to understand this is with the geometric series
$$f(x)=1+x+x^2+x^3+\cdots$$
This series clearly diverges if $|x|\gt1$. Yet it makes sense to say $f(2)=-1$, because of the identity
$$1+x+x^2+x^3+\cdots={1\over1-x}$$
which holds for all $|x|\lt1$. The right hand side is clearly meaningful for all $x\not=1$; it provides what's called an analytic continuation of the function $f$.
Something similar (but more complicated) occurs with the zeta function: The series definition $\zeta(s)=\sum n^{-s}$ makes no sense when the real part of $s$ is less than $1$, but there are identities with expressions that do. One of the things to look forward to, when you take a course in complex analysis, is learning why, for functions like the geometric series and the zeta function, you don't get conflicting identities that give different values for the analytic continuation, which is why it makes sense to talk about "the" zeta function $\zeta(s)$.
