The Riemann Hypothesis
Prime numbers have upset humanity for the last 2,300 years, ever since Euclid has proven that there was an infinity. Euclid proved this by simple proving that it could not be otherwise. Let's have a brief look at his proof. Any discussion on the prime numbers theorem and its consequences must start with a proof of the fact that indeed there is an infinity of prime numbers.
Euclid's proof starts with believing the contrary is true. Let us thus believe that there is a finite number of prime number: p1, p2,...pn. If this is so, then take a natural number N, that can be expressed as
N = p1*p2*...*pn + 1
It is obvious that this is not a multiple of any p1 through pn, as we have ...
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that there was an unlimited number of prime numbers, mathematicians tried in the following century to study how many primes there were less than a number x. This question was the basis for the Prime Number Theorem.
Let x be a positive real number and let us define ? (x) as the number of primes less or than equal to x. ? (x) is a function and will be studied as such. For small values of x, we can intuit the values for ? (x). Thus, for example, if x is 10, then the primes less then 10 are 2, 3, 5 and 7, so that ? (10) is 4. Similarly, ? (24) = 9. Our concern is for very large values of x.
As we have introduced the function ? (x), we are now able to state the Prime Number Theorem. Thus, this states that the number of primes not exceeding x is asymptotic to x/ln x (ln is the natural logarithm). Let us go back into history to discover how one has arrived at this theorem and how further proofs enriched its study. This does not help us too much, so we should give an ...
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limit had to be 1. Following the properties of factorials, he was the first to show that for sufficiently large numbers,
B < ? (x)/(x/ln x) < 6B/5, where B = 0.92129 and 6B/5 = 1.10555
It was Bernhardt Riemann in 1859 that continued Euler's work on series of numbers and made fundamental new discoveries within the prime numbers domain. Let us first start with a short description of Euler's work in the matter. Euler introduced in 1737 the so called product formula that states that for every s > 1, where s is a real number and the product runs through all the primes, we have
[pic] (1)
This was obtained by writing each of the factors on the right hand-side as an infinite ...
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"The Riemann Hypothesis." Essayworld.com. March 30, 2016. Accessed November 22, 2024. http://www.essayworld.com/essays/The-Riemann-Hypothesis/105518.
"The Riemann Hypothesis." Essayworld.com. March 30, 2016. Accessed November 22, 2024. http://www.essayworld.com/essays/The-Riemann-Hypothesis/105518.
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