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Bernhard Riemann (Reimann Hypothesis)

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ganalon

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Bernhard Riemann (Reimann Hypothesis)

PostTue Feb 26, 2013 5:16 pm

I am re-reading the book The Music of the Primes by Marcus du Sautoy. It is an extremely well written exploration of the search for ways of understanding prime numbers. In some ways prime numbers are very simple, in others they participate in the most difficult unproven mathematical problems of today.

du Sautoy does an excellent job of working up to the difficult problem called the Riemann Hypothesis. He also gives insight into the Riemann himself. For a short (1-hour) overview of the book from du Sautoy himself, check out



Riemann was a genius. Luckily one does not need to be a genius to understand significant parts of his work. It does take time, though and math smarts help too Riemann's work on the zeta function and its application to prime number theory can take weeks, months, even years to understand. If you can learn, discuss, and pass an oral "test" on the broad subject of Riemann's work with the zeta function, it seems likely that you could earn an undergraduate math scholarship and/or an improved chance of getting into graduate school for an advanced math degree. By the way, if you can prove the Riemann Hypothesis, you can win 1,000,000 dollars: http://www.claymath.org/millennium/

My interest in the subject stems from both curiosity and as a mental exercise that keeps my mathematical mind sharp. It's like taking your brain to the gym.

It has been known for over 2000 years that all positive integers greater than 1 (2,3,4,5,6...) can be constructed by multiplying primes together. So 6 can be written as 2x3, 12 as 2x2x3, 15 as 3x5. You get the idea. Primes can be written as 1 times itself. So 5 is also 1x5. Pretty straight forward? It can also be easily proven -- I'll let you Google it. Oh, and each number can be uniquely (also called canonically) expressed as a product of primes.

Here's my Riemann thought for today. I wanted to come up with concise notation. Here's what I came up with:

Image

See what I did? I made something simple look scary ! If you can get past any math anxiety, this equation becomes elegant -- at least more elegant than all the words it took to describe the concept two paragraphs before.

Let me explain the equation. x is any positive integer (1,2,3,4,5,...). pi is the ith prime number (2,3,5,7,..). M is short-hand for the product of primes. M is a matrix which for x=6 looks like [1,1]. This is like saying 6 = 2x3. This works because 2 is the 1st prime and 3 is the 2nd. The matrix [1,1] says take 2x3 to make the number 6.

Perhaps a list of examples will help:
1: [0]
2: [1]
3: [0,1]
4: [2]
5: [0,0,1]
6: [1,1]
7: [0,0,0,1]
8: [3]
9: [0,2]

For 8, M = [3]. This means, multiply the first prime, 2, together 3 times, or 2x2x2. For 9, M = [0,2], so multiply the second prime 3 together twice, as in 3x3. Not tough, right?

1 is bit odd at first. "Multiply the first prime, 2, times itself 0 times?" Actually, it means take 2 to the zeroth power. It turns out that "any number" to the 0th power is simply 1. So 1 = 20 = 1. It works! In other words, 0's in M can be replaced by 1 in the multiplication. Thus for 3, M = [0,1], so 3 = 1x3. For 5, M = [0,0,1], so 5 = 1x1x5.

The zeros of M can be essentially ignored, except as "counters". That is where the "i" in the equation comes in. It is simply there to count the number for each entry in the matrix, starting with 1. So for M=[1,2], the first pair (n,i) is (2,1), then i increments and the next pair (n,i) is (3,2). So x = 2 times 32 = 18.

Well, that's my introduction to Riemann. I look forward to comments and questions from math beginners and experts both!

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