Rebellion #1: The Natural Logarithm of Zero

This blog will become more serious progressing to the actual math of the proof. Until now, it’s been about introducing people of varying backgrounds to the idea of the Riemann Hypothesis. While there are other forms of the Riemann Zeta Function that will be talked about later, what will be fundamental to the proof is the ability to calculate the \ln{0} as a limit. Now that seems obvious, but I’m sure what I’m about to state here will start some discussion. That’s a good thing.


I’ve seen many people calculate the \lim_{m \to 0} \ln{m} as -\infty with no imaginary term since it is discontinuous depending on whether approaching from the left or right side. Inspecting limits of those forms, typically look something like:

\lim_{m \to 0^+} \ln{m} = -\infty

When calculating the left-side limit imaginary terms become introduced to handle the negative.

\lim_{m \to 0^-} \ln{m} = \ln{-1*|m|} = \ln{e^{\pi{i}}*|m|} = \ln{e^{\pi{i}}} + \ln{|m|} = \pi{i} -\infty

Now these aren’t exactly incorrect, but they ignore the fact that both sides of the limit have multiple values. Multiplying anything be -1^{2k} where k is a positive or negative integer doesn’t change value. but in this case it offers us a lot of insight. Reworking both sides, the limits develop as follows:

\lim_{m \to 0^+} \ln{-1^{2k}*m} = \lim_{m \to 0^+} \ln{e^{2k\pi{i}}*m} = \lim_{m \to 0^+} \ln{e^{2k\pi{i}}} + \ln{m} = 2k\pi{i} -\infty

\lim_{m \to 0^-} \ln{-1^{2k}*-1*|m|} =\lim_{m \to 0^-} \ln{e^{\pi{i}*(2k + 1)}*|m|} =\lim_{m \to 0^-} \ln{e^{\pi{i}*(2k + 1)}} + \ln{|m|} = (2k+1)\pi{i} -\infty

To look for continuity, the left-side limit is set equal to the right.

(2k+1)\pi{i} - \infty = 2k\pi{i} - \infty

While infinities don’t typically cancel, in this case they both are a representation of Re\{\ln{0}\} and thus are the same value. Another way of thinking about infinity in this case, which also fits well with this proof, is that of the largest possible integer. Since zeta functions and alternate representations of it are the sum of a series of values, the number of terms in the series shares this same representation. Anyway, removing the real infinities and looking only at the imaginary terms of the equation, it continues

(2k +1)\pi{i} = 2k\pi{i}

2k+1 = 2k

At this point it looks as if there is no continuity as this statement is not true. However, k is defined as any integer. Rearranging allows another limit to be taken.

\lim_{k \to \infty} \frac{2k + 1}{2k} = 1

Taking the \lim_{k \to \infty} or \lim_{k \to -\infty} the above statement holds true.

\lim_{m \to 0} \ln{m} = -\infty -\infty\pi{i} = -\infty + \infty\pi{i}

This reveals the fact that the ln{0}, or at least the limit of it, is actually dual-valued. The \pi term is left in for syntax purposes. It should be noted that again, this is a representation of infinity representing the largest possible integer.

This representation of \ln{0} will be vital to the upcoming proof and it cannot be completed without it.

The New Riemann Hypothesis, Part 2

Where Part 1 left off showed an alternate form of the Riemann Zeta Function is

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{e^{\omega_n{i}}}{n^\alpha}

This is not the final form of the equation. The next step is applying Euler’s formula. Euler came up with a proof to show that any imaginary exponent is a rotation in the complex plane. While I could take a massive detour here and explain to you exactly what Euler’s formula is and how to interpret and understand it, Kalid over at betterexplained.com has done a great job with it already. If you want to learn more about it, click here.

The mathematical statement of Euler’s law is this:

e^{xi} = \cos{x} + i\sin{x}

Taking Euler’s formula and applying x = \omega_n as seen in the numerator of the Zeta Function the following form is found.

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{\cos{\omega_n} + i\sin{\omega_n}}{n^\alpha}

Interpret this as a spiral placement of marbles on a balancing board as previously mentioned in the last post. How though? \cos{\omega_n} is an entirely real term. It determines how much of the distance from the center is in the real direction. The {i}\sin{\omega_n} term is purely imaginary because of the {i} being multiplied. It determines how much of the distance from the center is in the imaginary direction. \omega_n = \beta\ln{n}, which calculates the angle between the horizontal and a line drawn between each point created for each value of n. 1/{n^\alpha} represents the distance from the center. That’s how the interpretation in the image below is found.

Riemann Spiral Explained.001

Since \cos{\omega_n} is entirely real while i\sin{\omega_n} is entirely imaginary, it’s possible to split the Riemann Zeta Function into two different sums. Those sums would look like this

\sum \limits_{n=1}^\infty \frac{\cos{\omega_n}}{n^\alpha} = 0

\sum \limits_{n=1}^\infty \frac{i\sin{\omega_n}}{n^\alpha} = 0

It should be noted that i is in every term in the second sum, which means it can be factored out. After that it can be divided out since the sum is equal to zero. That would look something like this is mathematical notation.

\sum \limits_{n=1}^\infty \frac{i\sin{\omega_n}}{n^\alpha} = i\sum \limits_{n=1}^\infty \frac{\sin{\omega_n}}{n^\alpha}=0

Dividing through by i yields

\sum \limits_{n=1}^\infty \frac{\sin{\omega_n}}{n^\alpha}=0

From here, both the \sin{} terms and the \cos{} terms are treated as separately or put back together. For brevity’s sake, I will use the \sin{} terms only to show how the proof unfolds, then I will show how it is extended to both sets.

I’m not sure exactly how many posts are left in this blog until the proof in these posts are explained and the more formal proof is posted at the end, but if I ventured a guess, I would say it’s near the halfway point. I have to explain a few things about reducing the number of terms, the nature of calculating \ln{0} and adding in something which provides the necessary conditions for calculating \alpha. I look forward to sharing with those with you soon. Thanks for reading this far!

The New Riemann Hypothesis, Part 1

Focus is a strange thing. Too far away and things blend together, too close and we can’t see the picture. Sometimes it’s necessary to look at things from multiple distances to get a better understanding.


While others may use different variable names then I have used, the statements are all equivalent when stating the Riemann Hypothesis. The Riemann Hypothesis says for The Riemann Zeta Function to equal zero, alpha must equal \frac{1}{2} if it is between but not equal to 0 or 1. In mathematical terms it looks like this:

\zeta(s) = \sum\limits_{n=1}^\infty \frac{1}{n^s}=0   only when
s = \frac{1}{2} + \beta{i}    for    0< \alpha<1

If you stick with me for a bit, I’ll explain this in a physical sense.

Originally, in my proof for The Riemann Hypothesis, I thought of it as a statement about “wave heights” adding to zero. To put a picture into your head, the blue lines in the image below shows where the combinations of these “wave heights” were calculated. I thought The Riemann Hypothesis was a special statement about these “wave heights” adding to zero for certain values of s. This was a bad representation. I went through a couple dozen before I settled. Life lesson, question everything, even something you see. You may have the wrong perspective!

 

An easier to understand interpretation of the Riemann Hypothesis would be a set of points in a spiral-like pattern. Imagine each of the points being a marble sitting on a rectangular board. While doing this, take some liberties with your imagination because there would be infinite marbles. Placing the center of that board on a pin, if the board stayed balanced, it would represent The Riemann Zeta Function equaling zero. In proper math, that looks like this:

\zeta(s) = 0.

How does this relate to the variable s?

The variable s describes each marble’s placement with relationship to the center of the board. The real part of s tells us how far away from the center of the board, while the imaginary term tells us the angle that distance is at with respect to the horizontal axis. The image below may explain it.

Riemann Spiral Explained.001

Again, this board remaining perfectly balanced would represent The Riemann Zeta Function equaling zero, so the Riemann Hypothesis says that this board won’t balance unless each marble is a distance of \sqrt{Number Of Marble Placement} away from the center. Even in that case, there are only certain values of the angles between each marble placement that cause the board to balance, this would be determined by the imaginary part of s.

Disclaimer: We’re about to get into some math.

This math comes from my formal proof, which will be posted at the end of this blog series. Restating the Riemann Zeta Function and the definition of s:

\zeta(s) = \sum \limits_{n=1}^\infty \frac{1}{n^s} = 0
s= \alpha + \beta{i}

Replacing s in the Riemann Zeta Function with \alpha + \beta{i}. This is an equivalent substitution.

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{1}{n^{\alpha + \beta{i}}} = 0

From here, apply the laws of exponentiation that state equivalent definitions for manipulating variables with exponents. Two steps to illustrate each law being applied.

First Step

 \zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{1}{n^\alpha{n^{\beta{i}}}} = 0

Second Step

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{n^{-\beta{i}}}{n^\alpha} = 0

In this case, the first step was expanding the definition of the exponent, the second step was moving the imaginary portion to the numerator. More substitutions are coming. First, by the definition of natural logarithms, e^{\ln(n)} = n. Substituting this in for n in the numerator gives:

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{(e^{\ln{n}})^{-\beta{i}}}{n^\alpha}= \sum \limits_{n=1}^\infty \frac{e^{-\ln{n}\beta{i}}}{n^\alpha} = 0

Like all mathematicians, I don’t write more than necessary, so for simplicity sake, the substitution \omega_n = -\ln{n}*{\beta} is made.

\zeta(\alpha + \beta{i}) = \sum \limits_{n=1}^\infty \frac{e^{-\ln{n}\beta{i}}}{n^\alpha} = \sum \limits_{n=1}^\infty \frac{e^{\omega_n{i}}}{n^\alpha} = 0

This is almost the final form of my version of the Riemann Hypothesis, but it’s time to stop for now. I’ll finish this up in my next post and show you how this form comes back to the spiral form I showed at the beginning.

The Complex Variable s

Does something imaginary have no value?

If I tried to sell you an air guitar, you wouldn’t buy it. Unless I REALLY rocked it in a demonstration! Just Kidding.

Does that mean the air guitar has no value? The Air Guitar World Championships entertains tons of people, they even sell tickets to see it. This is just one example where something imaginary has no value itself, but value can be derived from it.


 

It isn’t the difficulty to understand that makes the variable, s, complex; it’s that it’s a combination of a real number and an imaginary number. If you don’t know what an imaginary number is click here.

In the Riemann Zeta Function, \zeta(s) = \sum\limits_{n=1}^\infty \frac{1}{n^s}, s is a variable that represents a complex number. It’a also the input to the Zeta Function. The variable s, as it relates to the Riemann Hypothesis is defined:

s = \alpha + \beta{i}

In this case, alpha (\alpha) is defined as being only a real number, which are numbers you know like 1,2,3.141598,-5, etc., while beta (\beta) times the imaginary number, i, represents the imaginary part of the complex number, or essentially a square root of a negative number that doesn’t have a physical interpretation, but arises often in physics.

Using s as a placeholder in the Zeta Function prevents having to write alpha and beta over and over again. The Riemann Hypothesis is a statement about alpha (\alpha). It says that under certain conditions, which we’ll talk about soon, the only way \zeta(s)=0 is if \alpha=\frac{1}{2}.

Time to visualize!

Imagine we have a race taking part on a street. The winner is the first person to run 1 mile down a straight path. This distance is measured in our alpha direction in the image below. Now imagine at the same time the races starts, an earthquake happens. For official reasons, the race has to continue, and can’t be reran. Due to the tremors, the contestants can’t seem to run straight anymore. As a result, they are veering off the straight-line and end up running more than a mile, but still the race is judged on the straight mile and not the wobbly one. We’ll call their fluctuations from the straight-line, our imaginary beta term. At the end of the race, runners were 1 mile from that start in the alpha direction, but were also some distance off the path, their actual position is similar to the variable s.

Running Path

So what’s the point of an imaginary number?

In this example, while only the time required to run the real distance, alpha matters for winning the race, it doesn’t necessarily mean that the winner was running at the fastest pace. For that we would need to know the length of the path each runner took divided by their finish time. If a faster runner had a longer path because of extra deviation from the road by the earthquake, then they might not win. Without tracking the fluctuation, beta in this case, it’s impossible to calculate the actual distance ran, or running speeds. Imaginary numbers are important because often times they are used to calculate other additional items.

To wrap up, the imaginary part of s, $\beta{i}&s=1$, produces valuable changes in the Zeta Function, but it in itself is not of value to the hypothesis statement which we’ll see in the next chapter. If you’re not clear, don’t worry. The variable s is just a definition describing real (\alpha) and imaginary (\beta{i}) terms.

The Riemann Zeta Function

It’s easy to believe that everything can be divided into clear cut categories, juxtaposed against each other, creating beautiful symmetry. To believe this, is the same as believing life is fair. It isn’t. And it never will be.

Mathematics isn’t the mass murderer of creativity it’s made out to be. In fact, it takes an artist to see its true form.


Riemann? What is that?

It’s not a what, it’s a who. Bernhard Riemann was a German Mathematician who made Herculean contributions to Mathematics, even if he did leave us with his mind-boggling hypothesis.

Zeta? What is that…greek?

Sure is. It’s a greek letter. It’s used as a unique symbol in The Riemann Hypothesis to represent the equation of interest.

Function? I feel like I remember that from math class.

You do. It’s an equation that relates an input to an output. Put in revenue and get out profit, or put in exercise and get out weight loss, it’s about creating a relationship between numbers.

The Riemann Zeta Function is an equation Bernhard Riemann developed and used the Greek letter Zeta to identify it. The input to the Riemann Zeta Function is the variable s, which we’ll talk about later. The full equation is shown below.

\zeta(s) = \sum\limits_{n=1}^\infty \frac{1}{n^s}

This equation starts with sigma (for the readers who could brush up on their greek letters – the abstract E)  which represents a sum of a number of mathematical terms based on the expression that follows it, in this case, the \frac{1}{n^s} . The number at the bottom of the sigma symbol tells us the starting number and the number on top tells us the number of terms to sum. For the Riemann Zeta Function, the output would be the total sum of an infinite number of terms, an example would be

\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...

all the way to infinity, which is seen as the symbol \infty. For the Zeta function, s is an exponent.


With all the facts and logic so far, I’m sure I’ve driven you away from the point I was trying to make…that Mathematics requires creativity. So a question for you,

How can you add up an infinite number of things?

If you can’t imagine how to do this, you need more imagination. Thanks for making my point!

Adding a never-ending stream of terms is a strange concept, but at times it’s possible. Let me give an example.

What do you see when you look at the equation below?

\sum\limits_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+...= \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...

It forms a pattern and we’re adding up the numbers in the pattern. Can you give the pattern a form? Could you create a drawing representing it? Could you sculpt something to describe it? Could you create music that sounds like it? This is where creativity comes in!

In this case, you can see that each term is 1/2 the size of the term before it. If you think of these terms as filling up a 1×1 square, as more terms are added, the entire square would be filled.

Screenshot 5:31:15 1:12 PM

This is only one form to imagine. I’d love to have people submit other forms that equation inspires in them. If you’re interested email me at infinit.brandon@gmail.com with a submission of your interpretation and a brief explanation, if I get enough unique submissions I’ll create a blog post showcasing them.

Since the entire area is 1, and each term added gets us closer to filling the entire square, as we approach an infinite number of terms, the value becomes 1. So the sum value can be written as follows:

\sum\limits_{n=1}^\infty \frac{1}{2^n} = 1

That example shows that infinite terms can sum to an actual value. I hope you find that interesting. If you don’t, then here is a fact, for some of these infinite sums that sum to finite values, a computer can never calculate them, it takes a person finding a form that is calculable. How’s that for countering the rise of the machines?

What is the Zeta Function doing?

Just as \sum\limits_{n=1}^\infty \frac{1}{2^n} can be given a form that can be visualized, the Riemann Zeta Function also has an interpretation. In fact, I’m sure there is an infinite number of interpretations and that’s the point. It’s people who can use their understanding of Mathematics and combine that with creativity who solve the unsolvable.

Proofs lacking creativity are pure exercises in Mathematics, strictly coherence to rules already established. Boring! Beyond boring, it leads us down long paths, not knowing where we’re heading but in a hurry to get there. Plus…no one wants to review them!

I want to explain what the Zeta Function is doing in an imaginative way, but a few details are still missing, like the variable, s, which I’ll cover soon. Until then, I’ll leave you with this, imagination is one way to come up with solutions to math problems, but more importantly it’s a way to find a way to state the problems themselves.

A Rebel and A Hypothesis

No person who was around when The Riemann Hypothesis was proposed is still alive today, in many cases their children and grandchildren aren’t either. Tons of proofs have been proposed, but even with a $1,000,000 prize, no one has put forth a proof that could be agreed upon as correct. Or maybe one was, but the author wasn’t convincing enough. It’s hard to say.

Just as Kerouac and Jobs thought, it takes a rebel to change things. I believe that The Riemann Hypothesis hasn’t been proven because people have been playing by the wrong “rules.” These “rules” have been widely accepted and even taught in schools. Say it ain’t so!

Rules allow us to “win.” But if winning meant bowling a 300 game and I defined 8 frames as a game instead of 10, you couldn’t do it. In the same manner, The Riemann Hypothesis can’t be proven if the “rules” don’t allow it to be.

I’ll change the “rules” and show you The Riemann Hypothesis always holds true.

Starting with this post as the first of a series, I’m going to develop a new form of The Riemann Hypothesis. I’ll develop equations and share insights that are intuitive. This blog is important for me because for a hypothesis to remain unproven through The Civil War, The Invention of the Automobile, The Invention of the Airplane and The Rise of Computers, there are only two possibilities:

  1. It’s actually not true, so it’s impossible to prove that it is, or
  2. It’s not understood.

Since computers have calculated trillions of cases that have so far shown The Riemann Hypothesis is true, #1 isn’t likely. That leaves us with the fact that people haven’t yet fully grasped what they are trying to solve.

I hate not understanding. Hopefully, you hate it too. This is one time where hate is for the good of society. This blog is my attempt to share my understanding and spread my hate…for lack of comprehension.

P.S. What would be useful for you to know before reading this blog series would be Laws of Exponentiation, Complex Numbers and Euler’s Formula but it’s not necessary, I will link to resources and include explanations along the way. Look for my next blog The Riemann Zeta Function next week!