en>FizykLJF: /* See also */ link added

2014-04-27T20:14:02Z

See also: link added

← Older revision		Revision as of 22:14, 27 April 2014
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	In [[number theory]], '''Niven's constant''', named after [[Ivan Niven]], is the largest exponent appearing in the [[prime factorization]] of any natural number ''n'' "on average". More precisely, if we define ''H''(1) = 1 and ''H''(''n'') = the largest exponent appearing in the unique prime factorization of a natural number ''n'' > 1, then Niven's constant is given by

	~~:<math>~~
	~~\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^n H(j) = 1+\sum_{k=2}^\infty \left(1-\frac{1}{\zeta(k)}\right)~~
	~~= 1.705211\dots \,~~
	~~</math>~~

	~~where ζ(~~''k'') is the ~~value~~ of the [[~~Riemann zeta function~~]] at the ~~point~~ ''~~k'' (Niven, 1969)~~.		If you're a performer, YouTube may help you [http://ganhandodinheironainternet.comoganhardinheiro101.com/ earn income] online. Make a brief video clip showcasing your talents. Are you a whiz with cosmetics? Make online tutorials that focus on makeup. Do you view yourself as a comedian? Why not film a brief comedy routine on hot news topics? When your videos have [http://www.comoganhardinheiro101.com/ferramentas/ ganhar dinheiro na internet] been uploaded to the site, advertisements will be attached to your page and this will lead to payment. There are many ways you can make money online; all you need is some basic information. This article should have helped you get started. Use these tips to earn as much or as little as you need online. <br><br><br>If you have the ability to type quickly, you may consider transcription jobs. 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	:<~~math~~>
	~~\sum_{j=1}^n h(j) = n + c\sqrt{n} + o (\sqrt{n}) \,~~
	<~~/math~~>

	~~where ''h''(1) = 1~~, ~~''h''(''n'') =~~ the ~~smallest exponent appearing in the unique prime factorization of each natural number~~ '~~'n'' > 1~~, '~~'o'' is~~ [~~[Big_O_notation#Little-o_notation\|little o notation]~~], ~~and~~ the ~~constant ''c''~~ is given by

	:<~~math~~>
	~~c = \frac{\zeta(\frac{3}{2})}{\zeta(3)}~~, \,
	~~</math>~~

	and ~~consequently~~ that

	: ~~<math> \lim_{n\~~to~~\infty} \frac{1}{n}\sum_{j=1}^n h(j) = 1~~. ~~</math>~~

	~~==References==~~
	*{{cite journal\|last = Niven\|first = Ivan M.~~\|authorlink = Ivan M~~. ~~Niven\|title = Averages of Exponents in Factoring Integers\|journal = Proceedings of~~ the ~~American Mathematical Society\|volume = 22\|issue = 2\|pages = 356–360\|date=August 1969\|doi = 10~~.~~2307/2037055\|jstor=2037055}}~~
	* Steven R. ~~Finch, ''Mathematical Constants'' (''Encyclopedia of Mathematics and its Applications''), Cambridge University Press, 2003~~

	~~==External links==~~
	* {{MathWorld\|urlname=NivensConstant\|title=Niven's Constant}}
	* {{SloanesRef\|sequencenumber=A033150}}

	~~[[Category:Mathematical constants]]~~
	~~[[Category:Number theory]]~~

en>OhGodItsSoAmazing: Added more nomenclature.

2013-09-19T17:57:16Z

Added more nomenclature.

← Older revision		Revision as of 19:57, 19 September 2013
Line 1:		Line 1:
	~~Friends call him Royal~~. ~~Some time ago he selected to live~~ in ~~Idaho. I am~~ a ~~cashier and I~~'~~ll be promoted soon~~. ~~The favorite hobby for my children and me~~ is ~~dancing and now I~~'~~m trying to earn cash with it~~.<br><br>~~Check out my web page ...~~ [~~http~~://~~www~~.~~Miketoerner~~.~~com/ActivityFeed/MyProfile/tabid/60/userId~~/~~35329/Default~~.~~aspx car warranty~~]		In [[number theory]], '''Niven's constant''', named after [[Ivan Niven]], is the largest exponent appearing in the [[prime factorization]] of any natural number ''n'' "on average". More precisely, if we define ''H''(1) = 1 and ''H''(''n'') = the largest exponent appearing in the unique prime factorization of a natural number ''n'' > 1, then Niven's constant is given by

			:<math>
			\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^n H(j) = 1+\sum_{k=2}^\infty \left(1-\frac{1}{\zeta(k)}\right)
			= 1.705211\dots \,
			</math>

			where ζ(''k'') is the value of the [[Riemann zeta function]] at the point ''k'' (Niven, 1969).

			In the same paper [[Ivan M. Niven\|Niven]] also proved that

			:<math>
			\sum_{j=1}^n h(j) = n + c\sqrt{n} + o (\sqrt{n}) \,
			</math>

			where ''h''(1) = 1, ''h''(''n'') = the smallest exponent appearing in the unique prime factorization of each natural number ''n'' > 1, ''o'' is [[Big_O_notation#Little-o_notation\|little o notation]], and the constant ''c'' is given by

			:<math>
			c = \frac{\zeta(\frac{3}{2})}{\zeta(3)}, \,
			</math>

			and consequently that

			: <math> \lim_{n\to\infty} \frac{1}{n}\sum_{j=1}^n h(j) = 1. </math>

			==References==
			*{{cite journal\|last = Niven\|first = Ivan M.\|authorlink = Ivan M. Niven\|title = Averages of Exponents in Factoring Integers\|journal = Proceedings of the American Mathematical Society\|volume = 22\|issue = 2\|pages = 356–360\|date=August 1969\|doi = 10.2307/2037055\|jstor=2037055}}
			* Steven R. Finch, ''Mathematical Constants'' (''Encyclopedia of Mathematics and its Applications''), Cambridge University Press, 2003

			==External links==
			* {{MathWorld\|urlname=NivensConstant\|title=Niven's Constant}}
			* {{SloanesRef\|sequencenumber=A033150}}

			[[Category:Mathematical constants]]
			[[Category:Number theory]]

en>Melcombe: Revert to revision 474253975 dated 2012-01-31 18:40:59 by Alph Bot using popups

2012-06-18T19:42:06Z

Revert to revision 474253975 dated 2012-01-31 18:40:59 by Alph Bot using popups

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Friends call him Royal. Some time ago he selected to live in Idaho. I am a cashier and I'll be promoted soon. The favorite hobby for my children and me is dancing and now I'm trying to earn cash with it.<br><br>Check out my web page ... [http://www.Miketoerner.com/ActivityFeed/MyProfile/tabid/60/userId/35329/Default.aspx car warranty]

Covariance function - Revision history

en>FizykLJF: /* See also */ link added

en>OhGodItsSoAmazing: Added more nomenclature.

en>Melcombe: Revert to revision 474253975 dated 2012-01-31 18:40:59 by Alph Bot using popups