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The 38 pages that link to Prime number theorem, each with the reason it gives.

• Prime numberRelated: It quantifies the long-term density of the primes whose infinitude Euclid proved.

• Bernhard RiemannRelated: Riemann’s analysis of zeta-function zeros helped establish the theorem’s deeper framework.

• Number theoryRelated: It gives a precise large-scale estimate for how frequently primes occur.

• Riemann zeta functionRelated: The zeta function’s pole at 1 and zero-free regions help establish the theorem.

• Prime-counting functionRelated: It gives the standard large-x approximation π(x) ≈ x/log x.

• Riemann hypothesisRelated: The zeta function’s zero-free region yields this theorem, while the hypothesis sharpens its error bound.

• Prime gapsRelated: It predicts the average gap near a large number is approximately its natural logarithm.

• Dirichlet seriesRelated: The analytic behavior of the zeta function’s Dirichlet series helps establish prime-distribution estimates.

• Euler productRelated: Analytic properties of the zeta function’s Euler product inform proofs about prime density.

• Möbius functionRelated: Bounds on summatory Möbius values became a major route toward this theorem.

• Sieve of EratosthenesRelated: The sieve generates the prime lists whose overall density the theorem describes.

• Bertrand's postulateRelated: It gives a broader asymptotic explanation for why primes occur in intervals like this.

• G. H. HardyRelated: Hardy and Littlewood developed methods for studying prime distribution beyond this foundational result.

• Exponential sumRelated: Exponential-sum estimates support analytic approaches to prime distribution.

• Green–Tao theoremRelated: It quantifies the primes' sparsity, the obstacle the proof must overcome.

• Euclid's theoremRelated: It quantifies how primes thin out, beyond establishing that they never end.

• Legendre's conjectureRelated: Its estimate predicts many primes near large squares but does not prove one in every such interval.

• Vinogradov's theoremRelated: Prime-distribution estimates underpin the exponential-sum bounds used in the proof.

• Euler–Mascheroni constantRelated: Mertens' product theorem features γ in the leading constant of a product over primes.

• John Edensor LittlewoodRelated: Littlewood gave a new proof and established that the error term changes sign infinitely often.

• Pafnuty ChebyshevRelated: Chebyshev’s bounds were an essential step toward its eventual proof.

• Brun's theoremRelated: It provides the wider historical problem of quantifying prime distribution.

• Bunyakovsky conjectureRelated: It represents the broader nineteenth-century drive to understand prime distribution.

• Montgomery pair correlation conjectureRelated: Explicit formulas connect zeta zeros with prime-counting errors, linking zero statistics to prime distribution.

• Bonse's inequalityRelated: It supplies the asymptotic perspective behind the growth of products of small primes.

• Erdős–Kac theoremRelated: Prime-counting estimates help determine the mean and variance scale, both about log log x.

• Riemann–von Mangoldt formulaRelated: The same zeta function links zero information to the distribution of primes.

• Brocard's conjectureRelated: Its broad density estimate gives context but does not establish the conjecture's local count.

• De Bruijn–Newman constantRelated: It is a classical consequence of zero-free information related to the zeta function, though weaker than the Riemann hypothesis.

• Divergence of the sum of the reciprocals of the primesRelated: It gives a precise distribution law that implies the reciprocal-prime sum diverges at logarithmic-logarithmic scale.