What formula does the video use to approximate prime density up to x?

Conditions

• x is a positive integer.
• The relation is an asymptotic approximation, not an exact equality for finite x.

Reasoning, step by step

1. Identify the natural logarithm of x, denoted $\ln (x)$.
2. Take the reciprocal of $\ln (x)$ to get $1/\ln (x)$.
3. Compare this theoretical curve with the empirical prime density curve.

Example

The video displays the formula #PRIMES / x ~ $1/\ln (x)$ and shows the green curve $y = 1/\ln (x)$ matching the red empirical density curve as x grows.

Common misconceptions

• Confusing the asymptotic approximation $1/\ln (x)$ with an exact formula for finite x.
• Assuming the tilde symbol (~) means exact equality rather than approximate or asymptotic agreement.

Watch the explanation

Explore next

• How is the Prime Number Theorem derived from this density formula?
• What numerical example checks the density approximation?
• Why is $1/\ln (x)$ related to the logarithmic spiral?