What is the Riemann Hypothesis? I've looked it up on Google but don't completely undestand it.
-MC
You wont understand it. It is related to the field of cryptology, and the use of prime numbers. Say I found a prime number with 54 digits, and used that number as a code overlay to my message to encyrpt it. I wouldn't want my enemy to know that secret 54 digit number, nor to be able to find it.
The prime number theorem allows one to figure the probablity to guessing the number, knowing how many digits it has. With the probability, one can figure out how long a super computer can iterate through all numbers around 54 digits, so it puts a limit on how that code can stand against the greatest supercomputer. One would like a time on the order of 3000 years to break the code, that is long enough.
The Reimann Hypothesis allows for the calculation of a "zeta" function value, which with the Prime number theorem, gives a good probability of finding the prime number.
Now if one is in the code breaking business, it has another importance.
Wow...
-MC
This hypothesis was used in one of the episodes of the TV show Numb3rs.
http://www.tv.com/numb3rs/episode/383244/summary.html
The Riemann Hypothesis is one of the most famous unsolved problems in mathematics, first formulated by the German mathematician Bernhard Riemann in 1859. It is about the distribution of prime numbers.
To understand the Riemann Hypothesis, we need to introduce the Riemann zeta function. The Riemann zeta function is defined as ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ..., where s is a complex number. This function is of great significance in number theory.
One of the main properties of the Riemann zeta function is its connection to the prime numbers. Bernhard Riemann discovered a relationship between the zeros of the zeta function and the distribution of prime numbers along the complex plane. The Riemann Hypothesis states that all nontrivial zeros of the Riemann zeta function lie on a specific vertical line in the complex plane, called the critical line. This critical line is defined by the formula Re(s) = 1/2, where Re(s) denotes the real part of the complex variable s.
To explain how to understand the Riemann Hypothesis, let's break it down into steps:
1. Start with the concept of prime numbers: Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, 2, 3, 5, 7 are prime numbers.
2. Familiarize yourself with the Riemann zeta function: The Riemann zeta function is a mathematical function defined as ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ..., where s is a complex number.
3. Understand the connection between the Riemann zeta function and prime numbers: Bernhard Riemann discovered a relationship between the zeros (the values of s for which ζ(s) = 0) of the zeta function and the distribution of prime numbers. The Riemann Hypothesis is the conjecture that all nontrivial zeros of the zeta function lie on the critical line Re(s) = 1/2.
4. Realize the significance of the critical line: The critical line is a specific vertical line in the complex plane that has a significant role in the Riemann Hypothesis. If the hypothesis is true, it means that we can predict the distribution of prime numbers more accurately.
It's important to note that the Riemann Hypothesis remains unproven, despite significant effort by mathematicians over the past century and a half. It has many profound implications for number theory and has connections to various mathematical fields. Many mathematicians consider it one of the most important unsolved problems in mathematics, due to its potential impact on prime numbers and the connections it has to other mathematical phenomena.