Prime Number Theorem and Its Expansion: A Puzzling Equation

In summary, the Prime Number Theorem is a fundamental result in number theory that estimates the distribution of prime numbers. It was first proven in the late 19th century using complex analysis techniques. The theorem has many applications in mathematics and has led to the development of other important theorems. The "Expansion" mentioned in the title refers to a refinement of the theorem that takes into account discrepancies for a more accurate approximation. The Prime Number Theorem is considered puzzling due to its simple equation accurately predicting the distribution of primes, and its connections to other areas of mathematics that are still not fully understood.
  • #1
autobot.d
68
0
Hi there,

working on Prime Number Theorem and the book gives an equality that I probably should know...

[itex]\frac{1}{log(2x)}= \frac{1}{logx}- \frac{log2}{log^{2}x} + O(\frac{1}{log^{3}x})[/itex]

and

[itex]\frac{1}{log^{2}2x} = \frac{1}{log^{2}x} + O(\frac{1}{log^{3}x}) [/itex]

Not sure what kind of expansion or what let's them draw this conclusion.
Any help is appreciated!
 
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  • #2
1/log2x = 1/(log2 + logx) = (1/logx)(1/(1 + {log2/logx})

1/(1 + u) = 1 - u + u2 - u3 + ...
Let u=log2/logx in the first line. x > 2 is condition.
 

FAQ: Prime Number Theorem and Its Expansion: A Puzzling Equation

What is the Prime Number Theorem?

The Prime Number Theorem is a fundamental result in number theory that gives an estimate for the distribution of prime numbers. It states that the number of primes less than a given number x is approximately equal to x/ln(x), where ln(x) is the natural logarithm of x.

How was the Prime Number Theorem proven?

The Prime Number Theorem was first proven in the late 19th century by two mathematicians, Jacques Hadamard and Charles Jean de la Vallée-Poussin. They used complex analysis techniques to prove the theorem.

What is the significance of the Prime Number Theorem?

The Prime Number Theorem has many important applications in mathematics, including in the study of prime numbers, number theory, and cryptography. It also provides insight into the distribution of primes and has led to the development of other important theorems, such as the Riemann Hypothesis.

What is the "Expansion" mentioned in the title of the equation?

The "Expansion" refers to a refinement of the Prime Number Theorem known as the Prime Number Theorem with Error Term. This expansion takes into account the discrepancies between the estimated number of primes and the actual number of primes, providing a more accurate approximation.

Why is the Prime Number Theorem a puzzling equation?

The Prime Number Theorem is considered puzzling because it is a simple equation that accurately predicts the distribution of prime numbers, even though the distribution of primes appears to be random. It also has connections to other areas of mathematics, such as complex analysis and the Riemann Hypothesis, which have yet to be fully understood.

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