# 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:

 $\pi(x)=\operatorname{li}x+R(x),$

where $\operatorname{li}$ is the logarithmic integral defined as

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

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. Narosa Publishing House, second edition, 1990. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0335.10001Zbl 0335.10001.
• 2 Harold Davenport. Markham Pub. Co., 1967. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0159.06303Zbl 0159.06303.
• 3 Paul Erdős. On a new method in elementary number theory. Proc. Nat. Acad. Sci. U.S.A., 35:374–384, 1949. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0034.31403Zbl 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. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=27.0154.01JFM 27.0154.01.
• 5 Donald J. Newman. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87:693–696, 1980. http://links.jstor.org/sici?sici=0002-9890%28198011%2987%3A9%3C693%3ASAPOTP%3E2.0.CO%3B2-UAvailable online at http://www.jstor.orgJSTOR.
• 6 Atle Selberg. An elementary proof of the prime number theorem. Ann. Math. (2), 50:305–311, 1949. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0036.30604Zbl 0036.30604.
• 7 Charles de la Vallée Poussin. Recherces analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxells, 1897.
 Title prime number theorem Canonical name PrimeNumberTheorem Date of creation 2013-03-22 11:45:18 Last modified on 2013-03-22 11:45:18 Owner bbukh (348) Last modified by bbukh (348) Numerical id 21 Author bbukh (348) Entry type Theorem Classification msc 11A41 Classification msc 55U10 Classification msc 55U30 Classification msc 55T25 Classification msc 55M05 Classification msc 55U15 Classification msc 81T25 Classification msc 81-XX Classification msc 20G42 Classification msc 81R50 Classification msc 17B37 Classification msc 81Q60 Classification msc 81V05 Classification msc 81T05 Classification msc 55R40 Defines pi(x) Defines logarithmic integral