# The Prime Number Theorem

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## Abstract

In the wake of Euclid’s proof of the infinitude of the primes, the question of how the primes were distributed among the integers became central — a question that has intrigued and challenged mathematicians ever since. The sieve of Eratosthenes provided a simple but very inefficient means of identifying which integers were prime, but attempts to find explicit, closed formulas for the nth prime, or for the number π(x) of primes less than or equal to a given number x, proved fruitless. Eventually extensive tables of integers and their least factors were compiled, detailed examination of which suggested that the apparently unpredictable occurrence of primes in the sequence of integers nonetheless exhibited some statistical regularity. In particular, in 1792 Euler asserted that for large values of x,  π(x) was approximately given by $$\dfrac{x} {\ln x}$$; six years later, Legendre suggested $$\dfrac{x} {\ln x - 1}$$ and (wrongly) $$\dfrac{x} {\ln x - 1.0836}$$ as better approximations; and in 1849, in a letter to his student Encke (translated in the appendix to Goldstein (1973)), Gauss mentioned his apparently long-held belief that the logarithmic integral

$$\displaystyle{ \text{li}(x) =\int _{ 2}^{x}\dfrac{1} {\ln t}\,\mathit{dt} }$$

gave a still better approximation. Using the notation f(x) ∼ g(x) to denote the equivalence relation defined by $$\lim _{x\rightarrow \infty }\dfrac{f(x)} {g(x)} = 1$$, those conjectures may be expressed in asymptotic form by the statements

$$\displaystyle{ \pi (x) \sim \dfrac{x} {\ln x},\quad \pi (x) \sim \dfrac{x} {\ln x - 1},\quad \text{and}\quad \pi (x) \sim \text{li}(x). }$$
(PNT)

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## Notes

1. 1.

Some texts instead define $$\text{li}(x)$$ as $$\lim _{\epsilon \rightarrow 0}\big(\int _{0}^{1-\epsilon }1/\ln t\,\mathit{dt} +\int _{ 1+\epsilon }^{x}1/\ln t\,\mathit{dt}\big)$$, which adds a constant (approximately 1.04) to li(x) as defined above, but does not affect asymptotic arguments.

2. 2.

In part for those proofs, Selberg was awarded a Fields Medal in 1950 and Erdős the 1951 Cole Prize in Number Theory.

3. 3.

In fact, for any absolutely convergent series $$\sum _{n=1}^{\infty }f(n)$$ of non-zero terms, if f(n) is completely multiplicative and no f(n) = −1, then $$\sum _{n=1}^{\infty }f(n) =\prod _{p\,\mathit{prime}}(1 - f(p))^{-1}$$.

4. 4.

The result of applying the formula

$$\displaystyle{\sum _{2\leq n\leq x}a(n)g(n) =\Big [\sum _{n\leq x}a(n)\Big]g(x) -\int _{\,2}^{x}\Big[\sum _{ n\leq t}a(n)\Big]\,g^{{\prime}}(t)\,\mathit{dt},}$$

valid whenever a(1) = 0 and g(x) has a continuous derivative on [2, x].

5. 5.

A detailed commentary on developments stemming from Riemann’s classic paper is given in Edwards (1974).

6. 6.

In the other cases Hadamard used the identity $$1/z^{\mu } = \dfrac{(-1)^{\mu -1}} {\Gamma (\mu )} \, \dfrac{d^{\mu -1}} {\mathit{dz}^{\mu -1}}(1/z)$$, together with Cauchy’s integral theorem, to obtain the general formula

$$\displaystyle{ J_{\mu } = \dfrac{1} {2\pi i}\int _{a-\infty i}^{a+\infty i}\dfrac{x^{z}} {z^{\mu }} \,\mathit{dz} = \left \{\begin{array}{@{}l@{\quad }l@{}} 0 \quad &\mbox{ if x < 1,} \\ \dfrac{(\ln \,x)^{\mu -1}} {\Gamma (\mu )} \quad &\mbox{ if x > 1}. \end{array} \right. }$$
7. 7.

In a statement quoted on page 198 of Narkiewicz (2000), de la Vallée Poussin agreed, but nevertheless claimed priority for the proof of that result.

8. 8.

Cf. the discussion in Bateman and Diamond (1996), p. 736.

9. 9.

According to Bateman and Diamond (1996), p. 737, Landau was the first to prove the PNT without recourse to that functional equation.

10. 10.

$$\sum _{n=2}^{N}f(n) =\int _{ 1}^{N}f(x)\,\mathit{dx} +\int _{ 1}^{N}(x - [x])f^{{\prime}}(x)\,\mathit{dx}$$.

11. 11.

Discussed in detail in Narkiewicz (2000), pp. 298–302.

12. 12.

Ingham’s theorem may alternatively be stated in terms of the Mellin transform $$\int _{1}^{\infty }f(t)t^{-s}\,\mathit{dt}$$. See, e.g., Korevaar (1982) or Jameson (2003), pp. 124–129.

13. 13.

Whereby f(x) = O(g(x)) for $$x > x_{1} \geq x_{0}$$ means that f is eventually dominated by g, that is, that f and g are both defined for x > x 0, g(x) > 0 for x > x 0, and there is a constant K such that | f(x) | ≤ Kg(x) for all x > x 1.

14. 14.

For as noted in the preceding section, the Wiener-Ikehara Theorem implies that the Prime Number Theorem follows from the absence of zeroes of the ζ-function on the line Re s = 1, a fact that is also implied by the Prime Number Theorem. (See, for example, Diamond 1982, pp. 572–573.)

15. 15.

Regrettably, the interaction between Erdős and Selberg in this matter was a source of lasting bitterness between them. Goldfeld (2004) provides a balanced account of the dispute, based on primary sources. As noted there, the issue was not one of priority of discovery, but “arose over the question of whether a joint paper (on the entire proof) or separate papers (on each individual contribution) should appear”.

16. 16.

Levinson’s paper won the Mathematical Association of America’s Chauvenet Prize for exposition in 1971. Nevertheless, after reading it, the number theorist Harold Stark commented “Well, Norman tried, but the thing is as mysterious as ever.” (Quoted in Segal 2009, p. 99.)

17. 17.

The Möbius inversion formula by itself does not suffice to give the desired bound on R(x), and that, in Levinson’s opinion, accounts for “the long delay in the discovery of an elementary proof” of the Prime Number Theorem. (Levinson 1969, p. 235)

18. 18.

“Nous allons voir qu’en modifiant légèrement l’analyse de l’auteur on peut établir le même résultat en toute rigeur.”

19. 19.

Three other instances given in Jameson (2003) are $$\sum _{n=1}^{\infty }\dfrac{\mu (n)} {n} = 0$$, where μ denotes the Möbius function, $$\sum _{n=1}^{\infty } \dfrac{\mu (n)} {n^{1+\mathit{it}}} = \dfrac{1} {\zeta (1 + \mathit{it})}$$, and $$\sum _{n=1}^{\infty }\dfrac{(-1)^{\Omega (n)}} {n} = 0$$, where $$\Omega (n)$$ denotes the number of prime factors of n, each counted according to its multiplicity.

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