Riemann Hypothesis

Riemann Hypothesis 02

What is the Zeta Function?

In general, the Zeta Function has a lot to do with the distribution of the primes. You remember what a prime is: any positive Integer that can only be divided by itself and one without any remainder. They begin as 2, 3, 5, 7, 11, 13...

Mathematicians have been interested in primes since antiquity. The reason for this interest can easily be forgotten until you remember the Fundamental Theorem of Arithmetic. Primes are something like a fundamental number, where all the other numbers are simply products of primes: 2 x 2 = 4, hence, four is not as fundamental as two, 2 x 3 = 6, therefore six is not as fundamental as two and three, and so on.

Understanding primes at a deep level adds greater understanding of numbers and therefore a greater understanding of mathematics. And the fact that all the universe is built on mathematics, from music to the chemical reactions that bring about life to the depths of the cosmos. Mathematics is behind it all. In other words, understand mathematics and one can glimpse how God has done it all. It is this fundamental aspect that obsesses mathematicians, perhaps a drive to see the face of God...perhaps. And it has been this way since at least the ancient Greeks. Considering that the first evidence of counting comes from the time humans domesticated the dog, some 20,000 years ago, this obsession probably goes even further back than records can show.

So, what is there to know about the primes other than this?

Well, mainly, no one really knows exactly how or why they are distributed among the other numbers. Studying them on a number line gets you nowhere. The first two primes are right next to each other, two and three. The next prime, five, skips over four. The next prime, seven, skips over six. Then next prime, eleven, skips over eight, nine and ten. But then the next prime, thirteen, only skips over twelve. The rest gets even less predictable. One thing that is known about them is that there are an infinite number of them, but a less infinite number of them than the composites, 4, 6, 8, 9, 10.... And the fact that there is such a possibility of even having a less infinite number than a greater infinite number mystifies the entire study ever so the more. See, the infinite is not just the infinite. There are an infinite number of infinities.

This is both the study of Number Theory and the study of Analysis.

When Bernhard Riemann wrote his paper back in the 1800's, the paper that first presented the RH, it broke ground on a new study of mathematics: Analytical Number Theory.

Okay, so what about the Zeta Function? Where does this fit in?

It stems from the Prime Number Theorem (PNT).

But to understand the PNT, one needs to know two functions:

* pi (x)

* Li (x)

The pi (x) function is called the Prime Counting Function. As x increases by one, x = 1, x = 2, x = 3..., pi (x) increases by one only when x is prime.

TABLE 1 The Prime Counting Function

This is called a Step Function, where many values of pi (x) repeat and then suddenly step up, often without any apparent clue or explanation.

Rather simple, huh. Easy to understand the function, but difficult to understand why or when it steps up. A general prediction to some of its pattern can be found in the second function.

The Li (x) function is called the Log Integral Function. As x increases, Li (x) increases by the inverse of the natural logarithm of x.

The inverse of something just means that something is divided by one or raised to the power of minus one.

The natural logarithm of something is simply the inverse of something e is raised to the power of, where e is Euler's number (Euler is pronounced "oiler") equal to 2.71828182..., which is the base of natural logarithms. Engineers commonly take the log of numbers with a base 10, but mathematicians use base e. On your scientific calculator, "log" will give the logarithm with base 10, but perhaps that's only because engineers make the scientific calculators. There is nothing significant to the log base 10 in mathematics, only log base e. On your scientific calculator, use the "ln" button for the natural logarithm.

Thus, if one inverts the result of (1) above, one gets just x. In the same,

Okay, so for Li (x), one simply takes the natural logarithm of all the numbers and inverts them. Then all you need to do is add up all the values to infinity...uh, or as many as you have time for.

Now, the Prime Number Theorem is that if you add up the values of (3) for all arguments from zero to inifnity, the the sum begins to equal the Prime Counting Function, becoming more and more equal toward infinity. And this was proven in 1896, though well after the RH was presented by Riemann.

This is rather interesting actually and has captivated many mathematicians for more than a century now.

What the heck are Euler's e and logarithms doing hanging around primes numbers? Primes are Integers and e is an Irrational number.

But they are linked!

This is mathematically how the PNT looks.

Okay, so Euler was a great mathematician, one of the best. But don't ever pronounce his name as "yooler" at dinner parties or they'll laugh you out the door. It's pronounced "oiler." So, Euler was fascinated with this and began to study a way to link logarithms to prime numbers in other ways. This effort led him to the Zeta Function.

Now, there are many ways to calculate the values of a given function. We'll start with the simplest version of the Zeta Function.

And this function works wonderfully for any Real number (I'll explain Real numbers soon enough) of s > 1. Let s = 5, raise all denominator Integers to the power of 5 and add up all the values. At s = 5, zeta of s = 1.0369277.... Let s = pi, the Irrational number equal to 3.14159..., raise all the denominator Integers to the power of pi and add up all the values. At s = pi, zeta of s = 1.1762415..., and so on for any positive Real number.

Okay, well that says very little about primes of course. Actually, it says nothing about primes. But we are in the area of values to some power.

Now, Euler rearranged the entire function in (5) very beautifully to come up with another version of the Zeta Function. It was a matter multiplying both sides of the infinite series of (5) by primes, inverting, adding, subtracting and the like to come up with this...

It really is only a slightly more complicated way to get the same result for a positive Real s in the Zeta Function. But you see, Euler was studying primes, particularly the Prime Number Theorem. He used the Zeta Function for it and so now does just about everyone else. Riemann certainly did.

Okay, so this is the beginning of the Zeta Function. Those two equations above allow one to solve for zeta of s when s is a positive Real number. But what about negative, Imaginary and Complex numbers? Can all numbers be applied to the Zeta Function.

Yes, all numbers can be applied with the single exception of one. One has no place in Zeta. Sorry, One.

In fact, it is all the other values of zeta of s that make it more interesting, as positive Real numbers always equal something close to one; the larger s gets, the closer to one zeta of s gets. That's not terribly interesting and obviously not why this problem has excited so many mathematicians over the years.

All the excitement lies on the Complex plane, which we will now begin to discuss.