Difference between revisions of "Prime Number Theorem"

Latest revision as of 02:41, August 16, 2015

The Prime Number Theorem is one of the most famous theorems in mathematics. It states that the number of primes not exceeding n is asymptotic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{n}{\log(n)}} , where log(n) is the logarithm of (n) to the base e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(n)\sim\frac{n}{\ln n}} .

The number of primes not exceeding n is commonly written as $\pi (n)$ , the prime counting function. An asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n). (This does not mean that a(n)-b(n) is small as n increases. It means the ratio of a(n) to b(n) approaches one as n increases.)

The Prime Number Theorem thus states that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(n) } ~Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n/ \log(n)} . That is, as n tends to infinity, the relative error between $\pi (n)$ and $n/\log(n)$ tends to zero. This can be expressed using limit notation as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1}

In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one. Put a third way, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n/log(n)} is a good approximation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(n)} .

Equivalent Statements

Carl Friedrich Gauss conjectured the equivalent statement that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(x)} was asymptotic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Li}(x)} defined as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Li}(x) = \int_2^x \frac{dt}{\ln t}} .

In fact, for large x this turns out to be a better approximation than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(x)} . The size of the error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Li}(x) - \pi(x) } is closely related to the behavior of the Riemann Zeta function

History of the Theorem

The theorem was first conjectured by Legendre and Gauss (independently) circa 1796. In 1848 and 1850, Chebyshev made significant progress towards proving the theorem using the Riemann zeta function and other non-elementary techniques. His non-elementary proof was completed in 1896 by Hadamard and Charles de la Vallee-Poussin.

The grand culmination of these efforts occurred in 1949 and 1950 when Paul Erdos and Atle Selberg presented an elementary proof of the theorem. This proof earned Selberg the highest prize in math, the Fields Medal.

@@ Line 1: / Line 1: @@
-The '''Prime Number Theorem''' is one of the most famous theorems in mathematics.  It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.
+The '''Prime Number Theorem''' is one of the most famous theorems in mathematics. It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.
-The number of primes not exceeding n is commonly written as <math>\pi(n)</math> the [[Prime counting function]], and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n).  (This does not mean that a(n)-b(n) is small as n increases.  It means the ratio of a(n) to b(n) approaches one as n increases.)
+:<math>\pi(n)\sim\frac{n}{\ln n}</math>.
-The Prime Number Theorem thus states that <math> \pi(n) </math>~<math> n/ \log(n)</math> .
+The number of primes not exceeding n is commonly written as <math>\pi(n)</math>, the [[prime counting function]]. An asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n).  (This does not mean that a(n)-b(n) is small as n increases.  It means the ratio of a(n) to b(n) approaches one as n increases.)
-In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one.  Put a third way, n/log(n) is a good approximation for <math>\pi(n)</math>.
+The Prime Number Theorem thus states that <math> \pi(n) </math>~<math> n/ \log(n)</math> . That is, as n tends to infinity, the [[relative error]] between  <math> \pi(n) </math> and <math> n/ \log(n)</math> tends to zero. This can be expressed using limit notation as
+:<math>\lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1</math>
+In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one.  Put a third way, <math>n/log(n)</math> is a good approximation for <math>\pi(n)</math>.
 == Equivalent Statements ==
@@ Line 19: / Line 23: @@
 The theorem was first conjectured by [[Legendre]] and [[Gauss]] (independently) circa 1796. In 1848 and 1850, [[Chebyshev]] made significant progress towards proving the theorem using the [[Riemann zeta function]] and other [[elementary techniques|non-elementary techniques]]. His non-elementary proof was completed in 1896 by [[Hadamard]] and Charles de la Vallee-Poussin.
-The grand culmination of these efforts occured in 1949 and 1950 when [[Paul Erdos]] and [[Atle Selberg]] presented an elementary proof of the theorem. This proof earned Selberg the highest prize in math, the [[Fields Medal]].
+The grand culmination of these efforts occurred in 1949 and 1950 when [[Paul Erdos]] and [[Atle Selberg]] presented an elementary proof of the theorem. This proof earned Selberg the highest prize in math, the [[Fields Medal]].
-[[Category:Mathematics]]
+[[Category:Number Theory]]