How is the natural logarithm related to the logarithmic spiral and prime density?

Conditions

• The spiral's rotation rate is modeled by $φ = \ln (r)$.
• Prime density is approximated by the reciprocal $1/\ln (x)$.

Reasoning, step by step

1. Define φ as rotations and r as distance for a logarithmic spiral.
2. Graph φ against r to find the natural logarithm relationship $φ = \ln (r)$.
3. Rewrite as $y = \ln (x)$.
4. Invert the function to $y = 1/\ln (x)$.
5. Match this inverted curve to the empirical prime density graph.

Example

The video shows a graph of the logarithmic spiral's properties following $y = \ln (x)$, then flips the y-axis to $1/\ln (x)$ to perfectly overlay the prime density curve.

Common misconceptions

• Assuming the logarithmic spiral directly counts primes; it models the density trend via the natural logarithm.
• Confusing the natural logarithm $\ln (x)$ with its reciprocal $1/\ln (x)$ in the context of prime density.

Watch the explanation

Explore next

• What formula approximates prime density for large x?
• Why does prime density decrease as x grows?
• How does the video derive the Prime Number Theorem from density?