Exercise · ZW-PRIME-NUMBER-THEOREM-010 · ★★★ Advanced
Let f(x) = π(x) and g(x) = x/ln x. If π(x) ~ g(x), which statement is true for large x?
Choose one option.
Show answer
- A. π(x)/g(x) tends to 1
Explanation
Asymptotic equivalence means the ratio tends to 1, not that the difference tends to 0. The difference may fail to tend to 0 even when the ratio does, and π(2x)/π(x) tends to 2, not 0.
Review the concept: Prime number theorem
The prime number theorem describes the asymptotic behavior of π(x), the number of primes less than or equal to x, as x tends to infinity.
\[\pi(x) \sim \frac{x}{\ln x}\]
More practice problems: prime number theorem
- ZW-PRIME-NUMBER-THEOREM-001 What does π(x) mean in the prime number theorem?
- ZW-PRIME-NUMBER-THEOREM-002 What does the theorem say about the behavior of π(x) as x tends to infinity?
- ZW-PRIME-NUMBER-THEOREM-003 Which symbol denotes that two functions are asymptotically equivalent?
- ZW-PRIME-NUMBER-THEOREM-004 According to the theorem, what does the ratio π(x)/(x/ln x) tend to as x grows?
- ZW-PRIME-NUMBER-THEOREM-005 What is the approximate value of π(100)?
- ZW-PRIME-NUMBER-THEOREM-006 If π(x) ~ x/ln x, what is the approximate estimate of π(1000) using x/ln x?
- ZW-PRIME-NUMBER-THEOREM-007 What is the approximate probability that an integer near x = 1000 is prime, according to the theore…
- ZW-PRIME-NUMBER-THEOREM-008 If p_n sim n ln n, what approximate estimate does the theorem give for the 100th prime?