Riemann Zeta Function By Hand

Question

This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?

Answer 1

The functional equation of the zeta function gives you the trivial zeros at once:

$$\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$$

if you input now $\,s=-2k\;,\;\;k\in\Bbb N\,$ , you get

$$\zeta(-2k)=2^{-2k}\pi^{-2k-1}\sin\left(\frac{\pi (-2k)}{2}\right)\Gamma(1+2k)\zeta(1+2k)=0\ldots$$

Question: Why in the negative even integers and not in the positive ones? Hint: check stuff about the gamma function.

Answer 2

Riemann did it - that's how he came up with his famous hypotheses. Of course, it was very arduous and he had to derive some pioneering results to do it, but that's why he is famous and I am not.

Do a search for "computing the Riemann zeta function" and you will get many hits. A good reference is Edwards' book "Riemann's Zeta Function".

Answer 3

Assuming you can calculate $n^s$ for natural $n$ and whatever $s$ you are interested in, you could try using:

$$\zeta(s)=\frac{\eta(s)}{1-2^{1-s}}$$

where

$$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^s},\quad\Re(s)>0$$

Define the partial sums of this to be

$$\eta(s,k)=\sum_{n=1}^k\frac{(-1)^{n+1}}{n^s}$$

While it is true that we have

$$\eta(s)=\lim_{k\to\infty}\underbrace{\eta(s,k)}_{\eta(s,k,0)}$$

we actually have the faster converging limit:

$$\eta(s)=\lim_{k\to\infty}\underbrace{\frac{\eta(s,k)+\eta(s,k+1)}2}_{\eta(s,k,1)}$$

Or even faster:

$$\eta(s)=\lim_{k\to\infty}\frac{\eta(s,k,1)+\eta(s,k+1,1)}2$$

Generally, you can define

$$\eta(s,k,n)=\begin{cases}\eta(s,k),&n=0\\\frac{\eta(s,k,n-1)+\eta(s,k+1,n-1)}2&,n>0\end{cases}$$

And then

$$\eta(s)=\lim_{k\to\infty}\eta(s,k,k)$$

converges fairly rapidly. The advantage here being that this approach converges for any complex $s$ and there is no difficult arithmetic involved.

---

After a few minutes of computation by hand, I approximated $\zeta(2)$ using only the first 6 terms of $\eta(2)$:

k | η(2,0,k)   | η(2,1,k)   | η(2,2,k)   | η(2,3,k)   | η(2,4,k)   | η(2,5,k)
------------------------------------------------------------------------------------
1 | 1.00000000 | 0.87500000 | 0.84027778 | 0.82899306 | 0.82498264 | 0.82347222
2 | 0.75000000 | 0.80555555 | 0.81770833 | 0.82097222 | 0.82196181
3 | 0.86111111 | 0.82986111 | 0.82423611 | 0.82295139
4 | 0.79861111 | 0.81861111 | 0.82166667
5 | 0.83861111 | 0.82472222
6 | 0.81083333

Compare this to the known value of $\eta(2)=\pi^2/12\approx0.82246703$