What is the formal limit definition of the Prime Number Theorem?

Conditions

• x is a positive integer approaching infinity.
• $π(x)$ is the actual number of primes less than or equal to x.

Reasoning, step by step

1. Identify the actual prime count $π(x)$.
2. Identify the estimate $x/\ln (x)$.
3. Form the ratio $π(x) / (x/\ln (x))$.
4. Take the limit as x approaches infinity.
5. Set the limit equal to 1.

Example

The video displays the equation lim_{x -> $\infty$} $π(x) / (x/\ln (x)) = 1$ and annotates it with labels 'As we look to infinity', 'actual count', 'estimate', and 'are equal'.

Common misconceptions

• Assuming the theorem states that $π(x)$ equals $x/\ln (x)$ exactly for all x.
• Misinterpreting the limit as applying to the difference rather than the ratio of the two functions.

Watch the explanation

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