More about Pi(x) and its derivatives

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In summary, Pi(x) is a function that counts the number of prime numbers less than or equal to a given number x, also known as the prime counting function. The derivatives of Pi(x) are known as the prime counting function derivatives and provide information about the distribution of prime numbers. The Riemann zeta function and the prime counting function are closely related through the Riemann hypothesis, which has important implications for the distribution of prime numbers. Pi(x) and its derivatives have significant applications in number theory, cryptography, physics, and other fields. While calculating these values becomes increasingly difficult for larger x, there are efficient algorithms and approximations that can be used, with the current record being for x = 10^26.
  • #1
eljose79
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studying the integral equation:

log(R(s)/s=Int(0,infinite)Pi(x)/x(x**s-1) and derivating and itegrating i have got to set an integral equation for dPi(x)/dx but now i wuld like to know if dPi(x)/dx could be expanded into a series of eigenfunctions of the kernel so we could solve it...in fact
dPi(n)/dn=(for big n)=1/l(x)-1/Ln(x)Ln(x) ubt i do not know if this will be enough.
 
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  • #2
I forgot to say the interval (2,infinite) in fact if it is not valid for dPi(x)/dx would be valid an expansion for d2Pi(x)/dx2 or higher?..thanks...
 
  • #3
pi is the prime counting function yes? well it only has one sided derivatives, and the derivative is everywhere 1
 

1. What is Pi(x)?

Pi(x) is a function that counts the number of prime numbers less than or equal to a given number x. It is also known as the prime counting function.

2. What are the derivatives of Pi(x)?

The derivatives of Pi(x) are known as the prime counting function derivatives. They are denoted by Pi'(x), Pi''(x), and so on. These derivatives provide information about the distribution of prime numbers and have important applications in number theory.

3. How is Pi(x) related to the Riemann zeta function?

The Riemann zeta function and the prime counting function are closely related through the Riemann hypothesis. The Riemann hypothesis states that the non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2. This has important implications for the distribution of prime numbers and is still an unsolved problem in mathematics.

4. What is the significance of Pi(x) and its derivatives?

Pi(x) and its derivatives play a crucial role in number theory and have important applications in cryptography, physics, and other fields. They provide insights into the distribution of prime numbers and are used in various algorithms and proofs.

5. Can Pi(x) and its derivatives be calculated for large values of x?

Calculating Pi(x) and its derivatives becomes increasingly difficult as x gets larger. However, there are efficient algorithms and approximations that can be used to calculate these values for very large x. In fact, the current record for calculating Pi(x) is for x = 10^26, using a combination of analytic and numerical methods.

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