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# prime number theorem

Define $\pi(x)$ as the number of primes less than or equal to $x$. The prime number theorem asserts that

$\pi(x)\sim\frac{x}{\log x}$ |

as $x\rightarrow\infty$, that is, $\pi(x)/\frac{x}{\log x}$ tends to 1 as $x$ increases. Here ${\log x}$ is the natural logarithm.

There is a sharper statement that is also known as the prime number theorem:

$\pi(x)=\li x+R(x),$ |

where $\li$ is the logarithmic integral defined as

$\li x=\int_{2}^{x}\frac{dt}{\log t}=\frac{x}{\log x}+\frac{1!x}{(\log x)^{2}}+% \cdots+\frac{(k-1)!x}{(\log x)^{{k}}}+O\left\{\frac{x}{(\log x)^{{k+1}}}\right% \},\qquad\text{for any fixed $k$}$ |

and $R(x)$ is the error term whose behavior is still not fully known. From the work of Korobov and Vinogradov on zeroes of Riemann zeta-function it is known that

$R(x)=O\left\{x\exp(-c(\theta)(\log x)^{\theta})\right\}$ |

for every $\theta>\tfrac{3}{5}$. The unproven Riemann hypothesis is equivalent to the statement that $R(x)=O(x^{{1/2}}\log x)$.

There exist a number of proofs of the prime number theorem. The original proofs by Hadamard [4] and de la Vallée Poussin[7] called on analysis of behavior of the Riemann zeta function $\zeta(s)$ near the line $\Re s=1$ to deduce the estimates for $R(x)$. For a long time it was an open problem to find an elementary proof of the prime number theorem (“elementary” meaning “not involving complex analysis”). Finally Erdős and Selberg [3, 6] found such a proof. Nowadays there are some very short proofs of the prime number theorem (for example, see [5]).

# References

- 1 Tom M. Apostol. Introduction to Analytic Number Theory. Narosa Publishing House, second edition, 1990. Zbl 0335.10001.
- 2 Harold Davenport. Multiplicative Number Theory. Markham Pub. Co., 1967. Zbl 0159.06303.
- 3 Paul Erdős. On a new method in elementary number theory. Proc. Nat. Acad. Sci. U.S.A., 35:374–384, 1949. Zbl 0034.31403.
- 4 Jacques Hadamard. Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France, 24:199–220. JFM 27.0154.01.
- 5 Donald J. Newman. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87:693–696, 1980. Available online at JSTOR.
- 6 Atle Selberg. An elementary proof of the prime number theorem. Ann. Math. (2), 50:305–311, 1949. Zbl 0036.30604.
- 7 Charles de la Vallée Poussin. Recherces analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxells, 1897.

## Mathematics Subject Classification

11A41*no label found*55U10

*no label found*55U30

*no label found*55T25

*no label found*55M05

*no label found*55U15

*no label found*81T25

*no label found*81-XX

*no label found*20G42

*no label found*81R50

*no label found*17B37

*no label found*81Q60

*no label found*81V05

*no label found*81T05

*no label found*55R40

*no label found*

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