Subsection 24.2.2 Motivating the Zeta function
The motivation for this definition comes from this function with the case \(s=1\text{.}\)
\begin{equation*}
\prod_{p\mid n}\left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots +\frac{1}{p^{e}}\right)=\sum_{d\mid n}\frac{1}{d}\text{.}
\end{equation*}
Try computing both sides of this and seeing how they come together for a few fairly composite \(n\text{,}\) like 12, 16, 18, 20, or 30.
Notice how every integer \(d\) formable by a product of the prime powers dividing \(n\) shows up precisely once (as a reciprocal) in the sum. This gives us a way into introducing limits.
What would happen if we introduced infinity in each term of the product, for instance?
\begin{equation*}
\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots\right)\left(1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\cdots\right)
\end{equation*}
By analogy, we should get a sum with exactly one copy of the reciprocal of each number divisible by only 2 and 3, e.g.
\begin{equation*}
\sum_{2\mid n\text{ or }3 \mid n}\frac{1}{n}\text{.}
\end{equation*}
There is no reason this wouldn’t continue to work for many prime factors.
Because every integer is
uniquely represented as a product of prime powers (
Fundamental Theorem of Arithmetic), this implies that we might multiply out the left-hand side of an
infinite product of
infinite sums to get
\begin{equation*}
\prod_{p}\left(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}+\cdots\right)=\sum_{n=1}^{\infty}\frac{1}{n}\text{.}
\end{equation*}
Since each of the multiplied terms on the left is an infinite geometric series, we can simplify the product slightly to write
\begin{equation*}
\prod_{p}\left(\frac{1}{1-1/p}\right)=\sum_{n=1}^{\infty}\frac{1}{n}\text{.}
\end{equation*}
Subsection 24.2.3 Being careful
So much for Euler’s contribution, a very impressive one. The only problem with all this is that both of these things clearly diverge!
Thus we cannot use a simple equality (\(=\)) for this discussion. Nonetheless, Euler’s intuition is spot on, and we will be able to fix this issue quite satisfactorily. For now, we can say is that, in some sense, the harmonic series is also an infinite product:
\begin{equation*}
\zeta(1)=\sum_{n=1}^{\infty}\frac{1}{n}\mathrel{\text{ “}\mathord{=}\text{” }}\prod_{p}\left(\frac{1}{1-1/p}\right)=\prod_{p}\left(\frac{1}{1-p^{-1}}\right)\text{.}
\end{equation*}
To make this rigorous, we should start talking about convergence. Recall this informal version of the integral test for series (see for example
Active Calculus).
Proposition 24.2.4. Integral test for series convergence.
Assume \(f\) is a positive decreasing function going to zero as \(x\to \infty\text{.}\) Then the series \(\sum_{i=1}^n f(i)\) converges if and only if the integral \(\int_1^{\infty}f(x)dx\) converges.
How does this apply to our situation? The improper integral in the case of \(\zeta(s)\) is
\begin{equation*}
\int_1^{\infty} x^{-s}\; dx\text{.}
\end{equation*}
As an example, in calculus one might have shown that \(\sum_{n=1}^\infty \frac{1}{n^2}\) converges by evaluating \(\int_1^\infty \frac{dx}{x^2}\text{.}\)
The general integral evaluates as
\begin{equation*}
\int_1^{\infty} x^{-s}\; dx=\frac{-x^{-s+1}}{1-s}\biggr|_1^{\infty}=\frac{1}{1-s}\left(1-\lim_{x\to\infty}\frac{1}{x^{s-1}}\right)\text{.}
\end{equation*}
For \(s\) a real number, this converges precisely when \(s>1\) (since that keeps \(x\) in the denominator), which begins to inform us about \(\zeta\text{.}\)
Fact 24.2.5.
The infinite sum \(\zeta(s)\) converges for all \(s>1\text{.}\)
But why is the (infinite) product equal to this infinite sum too? Is this product even meaningful? After all, it is not true in general that if a partial product equals a partial sum, then the ‘full’ sum is the ‘full’ product.
One has to carefully set up the convergence. If we can show that the product converges to the sum, then both will converge. Then it will make sense to say that
\begin{equation*}
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_p \left(\frac{1}{1-p^{-s}}\right)
\end{equation*}