Questions: Distribution of Primes and the Prime Number Theorem

Question 1

Near x = 10^6, approximately what fraction of integers are prime, according to the Prime Number Theorem?

• A. About 1 in 6
• B. About 1 in 14
• C. About 1 in 100
• D. About 1 in 1,000

Answer: (B)

The PNT says the density of primes near x is approximately 1/ln(x). Since ln(10^6) = 6 × ln(10) ≈ 6 × 2.303 ≈ 13.8, roughly 1 in every 14 integers near 10^6 is prime. This slow (logarithmic) rate of thinning is the quantitative content of the theorem — primes get rarer, but only as fast as 1/ln(x), not as fast as 1/x or 1/√x.

Question 2

The statement π(x) ~ x/ln(x) means which of the following?

• A. The error |π(x) − x/ln(x)| is bounded by a fixed constant for all x
• B. The ratio π(x) / (x/ln(x)) approaches 1 as x → ∞
• C. π(x) equals x/ln(x) exactly for all sufficiently large x
• D. x/ln(x) is always less than π(x)

Answer: (B)

The asymptotic notation f(x) ~ g(x) means the ratio f(x)/g(x) → 1 as x → ∞, not that the difference is small or that equality holds. The absolute error |π(x) − x/ln(x)| actually grows without bound; it is the relative error that vanishes. A better approximation is the logarithmic integral li(x), which has smaller relative error — but the PNT's claim is specifically about the ratio converging to 1.

Question 3

The average gap between consecutive primes near a large number x grows approximately like ln(x) as x increases.

• T. True
• F. False

Answer: True

This follows directly from the PNT. If primes have density ~1/ln(x) near x, then on average there is one prime per every ln(x) integers — meaning the average gap is ln(x). This gap grows without bound, confirming that primes thin out indefinitely, but at a slow logarithmic rate. Near x = 10^9, the average gap is about ln(10^9) ≈ 20.7.

Question 4

The proportion of integers that are prime approaches a nonzero constant as x → ∞.

• T. True
• F. False

Answer: False

The proportion of primes up to x is π(x)/x ~ 1/ln(x) → 0 as x → ∞. The density of primes goes to zero — they become increasingly sparse. However, there are still infinitely many primes (by Euclid's theorem), and their count π(x) grows without bound; it just grows more slowly than x itself. This is why the PNT is about the rate of thinning, not eventual disappearance.

Question 5

Why does the proof of the Prime Number Theorem involve the Riemann zeta function and complex analysis, rather than purely elementary reasoning about integers?

Model answer: The zeta function ζ(s) = Σ n^{−s} encodes the multiplicative structure of the integers via the Euler product ζ(s) = Π_p (1 − p^{−s})^{−1}, connecting it directly to the distribution of primes. The analytic behavior of ζ(s) — especially its zeros in the complex plane — controls how closely π(x) approximates x/ln(x). Showing ζ(s) has no zeros on the line Re(s) = 1 is what the 1896 proofs established, and this zero-free region drives the asymptotic. Complex analysis provides tools (contour integration, residue calculus) to extract precise asymptotic information from generating functions that purely arithmetic methods cannot match.

This is the founding example of analytic number theory: importing tools from complex analysis to answer questions about integer structure. The integers look disconnected and combinatorial, but encoding their multiplicative structure in a complex function reveals regularity through analytic behavior. The connection runs: prime distribution ↔ zeros of ζ(s) ↔ analytic behavior of a complex function — each step making the problem more tractable.