The Riemann Zeta Function English Language Essay

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The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day.

n 1739, Euler found the rational coefficients C in zeta(2n)=Cpi^(2n) in terms of the Bernoulli numbers. Which, when combined with the 1882 proof by Lindemann that pi is transcendental, effectively proves that zeta(2n) is transcendental. 

As defined above, the zeta function zeta(s) with s=sigma+it a complex number is defined for R[s]>1. 

On the real line with x>1, the Riemann zeta function can be defined by the integral

he Riemann zeta function satisfies the reflection functional equation

3~5 sentence summary of the article/chapter

This articles deals with not only the fundamental theories of Riemann zeta function but also very advanced theories that requires me to read thoroughly. Reading this article and citing the author will help my writing to be more reliable and the depth of my extended essay will be deepen.

Title: Riemann Hypothesis

Author: Goodman, Len and Weisstein, Eric W.

Website: http://mathworld.wolfram.com/RiemannHypothesis.html

Classification: background

Date the article was downloaded/found: 2013-02-24

Date the article was published: Unknown

5 important quotations and paraphrases

First published in Riemann's groundbreaking 1859 paper (Riemann 1859), the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros

A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated

The Riemann hypothesis has thus far resisted all attempts to prove it

Proof of the Riemann hypothesis is number 8 of Hilbert's problems and number 1 of Smale's problems.

3~5 sentence summary of the article/chapter

This article deals with the Riemann Hypothesis. Riemann Hypothesis is a hypothesis by Riemann that indirectly states the Riemann zeta function zeros. Even though it is not directly related to my topic citing this article will enhance the credibility and the depth of my Extended Essay will be deepen.

Article Title: Largest Prime Number Discovered

Author: Tia Ghose and LiveScience

Website: http://www.scientificamerican.com/article.cfm?id=largest-prime-number-disc

Classification: background

Date the article was downloaded/found: 2013-02-25

Date the article was published: 2013-02-05

5 important quotations and paraphrases

The largest prime number yet has been discovered - and it's 17,425,170 digits long. 

The new prime number crushes the last one discovered in 2008, which was a paltry 12,978,189 digits long.

By using the largest prime number, the most secure safe can be made

Great Internet Mersenne Prime Search (GIMPS) harnesses about 360,000 processors operating at 150 trillion calculations per second. This is the thirdprime number discovered by Cooper.

"If you were to do it that way it would take longer than the age of the universe," he said.

3~5 sentence summary of the article/chapter

This article is about the largest prime number found in earth. The prime numbers can be sorted out by the Riemann Theory and used in building the locks. If larger the prime number, the safer the lock will be. Therefore, by making the lock with the largest prime number found, there will be a magnificent lock which is not able to hack.

Article Title: Proof claimed for deep connection between prime numbers

Author: Philip Ball and Nature magazine

Website:http://www.scientificamerican.com/article.cfm?id=proof-claimed-for-deep-connection-between-prime-numbers

Classification: background

Date the article was found: 2013-02-25

Date the article was published: 2012-09-10

5 important quotations from the article

The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.

The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat's Last Theorem, but in some ways it is more significant.

It turns out that this conjecture encapsulates many other Diophantine problems, including Fermat's Last Theorem

Conrad says that the work "uses a huge number of insights that are going to take a long time to be digested by the community".

Mochizuki's track record certainly makes the effort worthwhile. "He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence," says Conrad. 

3~5 sentence summary of the article

This article deals with the proof claimed for deep connection between Prime Numbers. Mathematicians strived for decades to found the connection between the prime numbers. If this proof is reliable, the larger amount of prime numbers will easily be found and the lock on our safe will be more secure.

Article Title: Great Mathematical Problems

Author: Ian Stewart

Website: http://www.huffingtonpost.com/ian-stewart/great-mathematical-problems_b_2569381.html

Classification: background

Date the article was found: 2013-02-25

Date the article was published: 2013-01-28

5 important quotes from the article

Mathematicians have always been fascinated by the great problems of their subject, the deep and difficult questions that they would give their right arm to answer

Most great problems become great when something that looks as if it ought to be simple turns out to be much harder than anyone expected.

The Riemann Hypothesis, about a fundamental issue in complex analysis related to prime numbers, continues to baffle the world's mathematicians, and remains as impenetrable as ever after 150 years.

Great mathematical problems sometimes arise from questions about the natural world, but more often they emerge because gaps appear in our mathematical knowledge. 

New scientific discoveries often lead to new mathematics. Isaac Newton's laws of motion and gravity didn't provide an immediate understanding the solar system. 

3~5 sentence summary of the article

This article mainly discusses about the unsolved great mathematical problems in the universe. It was interesting to know that the Riemann Hypothesis was included in them. I could use this article to show that the Riemann Hypothesis took several thousand years to be proven.

Article Title: Calculating professor takes one small step closer to infinity

Author: Konstantin Kakaes

Website: http://www.theage.com.au/technology/technology-news/calculating-professor-takes-one-small-step-closer-to-infinity-20130207-2e12g.html

Classification: background

Date the article was founded: 2013-02-25

Date the article was published:2013-02-07

5 important quotes in the article

Euclid's proof that there is an infinite number of prime numbers is both one of the simplest mathematical proofs and one of the oldest.

Prime numbers - integers that are divisible only by themselves and 1 - are the easiest path into understanding both rigour and mysticism in mathematics.

The new number has 17,425,170 digits - just writing them down makes for a 22.45-megabyte text file.

Mersenne primes are numbers like 3=2 to the second minus 1 , 7=2 to the third minus 1 or 31=2 to the fifth minus 1 that are 1 less than a power of 2.

3~5 sentence summary of the article

This article mainly discusses about the professor who strives to calculate the larger prime number than it is known as the largest. He found a larger prime number and the new number had 17425170 digits.

Article Title: Spider Mathematics: Story and Theorem

Author: Ahn Jaechan

Website: Book

Classification: background

Date the article was founded: 2013-03-03

Date the article was published: 2004-10-05

5 important quotes in the article

An Ancient Greek mathematician Eratosthenes proved that the numbers of the prime numbers are unlimited

In BC 275, Euclid also proved that the numbers of the prime numbers are unlimited using the Mathematical Induction.

Goldbach's hypothesis is that even number higher than 2 can be restated by two prime number's summation.

How can prime numbers be used in reality?

The fundamental aspects of even number and odd number can be used to determine how to prove that there is no connection between the prime numbers

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