Nontrivial Zeros in Complex Numbers, the Formal Definition of the Riemann Hypothesis
Episode #6 of the course Great math problems for the 21st-century mind by John Robin
Yesterday, we saw that
1 + 2 + 3 + 4 + … = −1/12
and even weirder:
1 + 1/(2n) + 1/(3n) + 1/(4n) + 1/(5n) + … = 0
for all even negative values of n.
I only explored the case where n = −1, but similar fascinating results emerge for all the odd negative values:
n = −3, then 1 + 1/(2–3) + 1/(3–3) + 1/(4–3) + 1/(5–3) + … = 1 + 8 + 27 + 64 + 123 + …
n = −5, then 1 + 1/(2–5) + 1/(3–5) + 1/(4–5) + 1/(5–5) + … = 1 + 32 + 243 + 1024 + 3125 + …
These results follow from similar proofs to the one I demonstrated yesterday for n = −1. The pattern is not simple, but it does follow a formula using the Bernoulli Numbers.
Today, we’re going to look at where all this weirdness is leading us and exactly how it’s central to the Riemann Hypothesis.
Before we can go any further, we must now give our p-series a new name, the Riemann zeta function:
ζ(z) = 1 + 1/(2z) + 1/(3z) + 1/(4z) + 1/(5z) + …
Notice that I used z instead of n. I’ll talk about that in a moment.
Now, up to this point, we’ve been exploring the question, “What does 1 + 1/(2z) + 1/(3z) + 1/(4z) + 1/(5z) + … add to?” Or equivalently, “What are all the values of ζ(z)?”
Mathematicians have been exploring that question for a long time, but it was the more specific question, “What are all the values of z for which ζ(z)=0?”, that led to a hypothesis that is now worth $1,000,000 if someone can prove (or disprove) it. These values are commonly called the zeros of the Riemann zeta function.
Up to this point, we have only been exploring integer values of z. The values z = −2, −4, −6,—i.e., all even negative integers—are called trivial zeros. Mathematicians, however, are interested in the nontrivial (i.e., more interesting) zeros, and in order to find these, it means looking outside ordinary numbers.
The reason I switched to z is because searching for the nontrivial zeros means searching through the complex numbers.
You might be familiar with complex numbers, but if not, here’s a quick brush up.
What is √−1? Because there is no way two equal numbers can multiply to make a negative number—i.e., there is no actual real number answer for √−1—we simply let √−1 = i. We call i an imaginary number.
More generally, what is √−x where x is any real (i.e., decimal) number? We can simply write:
√−x = √(−1(x))= i√x
Now, it turns out that every number we use, like 1.467903, can be written in the form:
a + bi
In the case of i√x, we simply write:
0 + (√x)i
In the case of 1.467903, we simply write:
1.467903 + 0i
We call a+bi a complex number, where a and b are real numbers. When we try to visualize a complex number, we do so by making a plot much like the familiar (x,y) graph (where “Im” stands for “Imaginary axis” and “Re” for “real axis”):
Notice that “real axis” is just the ordinary (i.e., real) number line. You can think of every ordinary number—e.g., 1.467903—as having complex number extensions above and below vertically. For example, 1.467903 + i, 1.467903 + 2i, etc. lie above, and 1.467903 − i, 1.467903 − 2i, etc. lie below.
The Riemann zeta function then considers every possible complex number, z in ζ(z), and the Riemann Hypothesis speculates on where all the nontrivial zeros of the function will be.
How many will we find?
That’s exactly what mathematician Bernhard Riemann hypothesized about. He claimed there are infinitely many. But not just that: He claimed every one of them will only be found in the form z = 1/2 + bi.
More than 150 years after Riemann hypothesized this, mathematicians have still not been able to prove it. But they have found some important results.
G.H. Hardy, for example, proved that there are infinitely many zeros of the form 1/2 + bi. However, it remains to be shown that these are the only zeros. It’s this proof that is worth $1,000,000.
Tomorrow, I’ll move on to another one of my favorite problems—another behemoth worth $1,000,000.
If you want to learn more about the Riemann Hypothesis and further visualize what is happening with the beautiful (and strange) patterns in the zeta function, here’s one of my favorite videos on it.
And for further reference, in case you want to dig deeper and try your hand at it, this video by Numberphile explains some great theoretical background.
The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of our Time by Keith Devlin
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