Difference between revisions of "Prime Number Theorem"
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== Equivalent Statements == | == Equivalent Statements == | ||
| − | [[Gauss]] conjectured the equivalent statement that <math>\pi(x)</math> was asymptotic to <math>\mbox{Li}(x)</math> defined as: | + | [[Carl Friedrich Gauss]] conjectured the equivalent statement that <math>\pi(x)</math> was asymptotic to <math>\mbox{Li}(x)</math> defined as: |
<math> \mbox{Li}(x) = \int_2^x \frac{dt}{\ln t}</math>. | <math> \mbox{Li}(x) = \int_2^x \frac{dt}{\ln t}</math>. | ||
In fact, for large x this turns out to be a better approximation than <math>\pi(x)</math>. | In fact, for large x this turns out to be a better approximation than <math>\pi(x)</math>. | ||
Revision as of 21:24, March 14, 2007
The Prime Number Theorem is one of the most famous theorem 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.
The number of primes not exceeding n is commonly written 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 \pi(n)} , 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.)
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 n/ \log(n)} .
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 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)} .