How can the prime number theorem be used to understand this proof?

In summary, the conversation discusses a proof by William Miller that makes use of the prime number theorem. The proof involves the use of two functions, A(x) and theta(x), and an identity that relates theta(x) to the prime counting function, pi(t). The identity is derived from the asymptotic behavior of A(x) and theta(x) and the fact that pi(t) jumps by 1 when t is prime.
  • #1
~Death~
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could someone please help me understand this proof given in an article by William Miller

(attatched)

its supposed to follow from the prime number theorem that given,

A(x) which is the sum of all primes less than or equal to x

and theta(x) which is the sum of the log of all primes less than or equal to x

A(x) ~ x^2/(2logx) and theta(x) ~ x

the following identity is used, theta(x) = integral from 1 to x of log(t)d(pi(t))

where pi(t) is the prime counting function. I don't understand why this is.

Here ~ means asymptotic to i.e. lim n->infinity f(x)/g(x)=1
 

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  • #2
pi(t) jumps by 1 when t is prime. Therefore log(t) d(pi(t)) contributes log(t) for prime integers and 0 for all other values of t.
 

1. What is the prime number theorem?

The prime number theorem is a mathematical theorem that gives an estimate for the distribution of prime numbers. It states that the number of primes less than or equal to a given number n is approximately equal to n/ln(n), where ln(n) is the natural logarithm of n.

2. Who discovered the prime number theorem?

The prime number theorem was first conjectured by mathematician Adrien-Marie Legendre in 1797 and later proven by mathematician Carl Friedrich Gauss in 1798. However, it was independently rediscovered and proven by mathematicians Pierre-Simon Laplace and Johann Carl Friedrich Gauss in 1808.

3. What is the significance of the prime number theorem?

The prime number theorem is significant because it provides a deeper understanding of the distribution of prime numbers, which are the building blocks of all natural numbers. It also has important applications in fields such as cryptography and number theory.

4. Is the prime number theorem still an open problem?

No, the prime number theorem has been proven and is considered a fundamental result in number theory. However, there are still many open problems related to prime numbers, such as the Riemann hypothesis, which is closely connected to the prime number theorem.

5. Can the prime number theorem be generalized to other types of numbers?

Yes, the prime number theorem has been generalized to other types of numbers, such as the distribution of prime numbers in arithmetic progressions. There have also been attempts to generalize it to other types of numbers, such as Gaussian primes and elliptic curves, with varying degrees of success.

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