prime number theorem
PrimeNumberTheorem
2001-10-15 19:36:21
2006-09-05 13:39:13
Theorem
pi(x)
logarithmic integral
number theory
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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:
\begin{equation*}
\pi(x)=\li x+R(x),
\end{equation*}
where $\li$ is the logarithmic integral defined as
\begin{equation*}
\li x=\int_2^x \frac{dt}{\log t}=\frac{x}{\log x}+\frac{1!x}{(\log
x)^2}+\dotsb+\frac{(k-1)!x}{(\log x)^{k}}+O\left\{\frac{x}{(\log
x)^{k+1}}\right\},\qquad\text{for any fixed $k$}
\end{equation*}
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
\begin{equation*}
R(x)=O\left\{x \exp(-c(\theta)(\log x)^\theta)\right\}
\end{equation*}
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 \cite{cite:hadamard_pnt} and de~la
Vall\'ee~Poussin\cite{cite:poussin_pnt} 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 Erd\H{o}s and Selberg
\cite{cite:erdos_elempnt,cite:selberg_elempnt} found such a proof.
Nowadays there are some very short proofs of the prime number
theorem (for example, see~\cite{cite:newman_shortpnt}).
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\bibitem{cite:newman_shortpnt}
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