en>Freeze S: Preface cleaned up.

2014-11-12T14:44:11Z

Preface cleaned up.

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en>Mr. Granger: Repairing links to disambiguation pages - You can help!

2014-02-16T06:30:01Z

Repairing links to disambiguation pages - You can help!

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en>Noix07 at 14:08, 23 January 2014

2014-01-23T14:08:53Z

← Older revision		Revision as of 16:08, 23 January 2014
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	~~The other day I woke up~~ and ~~noticed - I have been single for some time today and after much intimidation from friends I today find myself registered~~ [~~http://www~~.~~banburycrossonline.com luke bryan cheap concert tickets~~] ~~for online dating. They guaranteed me that there are~~ a ~~lot of pleasant~~, ~~ordinary and interesting individuals to meet~~, ~~therefore~~ the ~~pitch is gone by here!~~<br>~~I endeavor to keep~~ as ~~physically find luke bryan; [http~~:/~~/www.hotelsedinburgh.org similar webpage], healthy as possible coming~~ to the ~~fitness center many times a week~~. ~~I enjoy my athletics~~ and ~~endeavor to~~ [~~http~~://~~www~~.~~museodecarruajes.org luke bryan concert dates~~] ~~perform or see as several a potential~~. ~~I shall frequently at Hawthorn fits being winter~~. ~~Notice: Supposing that you would considered buying a hobby I don't brain~~, ~~I've observed~~ the ~~carnage~~ of ~~fumbling matches at stocktake revenue~~.<br>~~My family and buddies are wonderful and hanging out together at tavern gigabytes or dishes~~ is ~~definitely vital~~. ~~As I come to realize that one can do not own~~ a ~~significant conversation vip tickets luke bryan~~ - [~~http:~~//~~lukebryantickets~~.~~iczmpbangladesh.org lukebryantickets.iczmpbangladesh.org], together with~~ the ~~sound I haven't ever been in to nightclubs. I likewise have 2 really cunning and~~ [~~http:~~//~~Www.Bing.com~~/~~search?q~~=~~unquestionably~~+~~cheeky&form~~=~~MSNNWS&mkt=en~~-~~us&pq=unquestionably+cheeky unquestionably cheeky] canines that are consistently excited to meet up fresh individuals~~.<br><br>~~Look into my weblog luke bryan~~ the ~~today show~~ ([~~http:~~//~~lukebryantickets.lazintechnologies.com lukebryantickets.lazintechnologies.com~~])		In [[mathematics]], the '''Bell series''' is a [[formal power series]] used to study properties of arithmetical functions. Bell series were introduced and developed by [[Eric Temple Bell]].

			Given an [[arithmetic function]] <math>f</math> and a [[Prime number\|prime]] <math>p</math>, define the formal power series <math>f_p(x)</math>, called the Bell series of <math>f</math> modulo <math>p</math> as:

			:<math>f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.</math>

			Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the ''uniqueness theorem''. Given [[multiplicative function]]s <math>f</math> and <math>g</math>, one has <math>f=g</math> [[if and only if]]:
			:<math>f_p(x)=g_p(x)</math> for all primes <math>p</math>.

			Two series may be multiplied (sometimes called the ''multiplication theorem''): For any two [[arithmetic function]]s <math>f</math> and <math>g</math>, let <math>h=f*g</math> be their [[Dirichlet convolution]]. Then for every prime <math>p</math>, one has:

			:<math>h_p(x)=f_p(x) g_p(x).\,</math>

			In particular, this makes it trivial to find the Bell series of a [[Dirichlet convolution\|Dirichlet inverse]].

			If <math>f</math> is [[completely multiplicative]], then:
			:<math>f_p(x)=\frac{1}{1-f(p)x}.</math>

			==Examples==

			The following is a table of the Bell series of well-known arithmetic functions.

			* The [[Möbius function]] <math>\mu</math> has <math>\mu_p(x)=1-x.</math>
			* [[Eulers phi function\|Euler's Totient]] <math>\varphi</math> has <math>\varphi_p(x)=\frac{1-x}{1-px}.</math>
			* The multiplicative identity of the [[Dirichlet convolution]] <math>\delta</math> has <math>\delta_p(x)=1.</math>
			* The [[Liouville function]] <math>\lambda</math> has <math>\lambda_p(x)=\frac{1}{1+x}.</math>
			* The power function Id<sub>k</sub> has <math>(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.</math> Here, Id<sub>k</sub> is the completely multiplicative function <math>\operatorname{Id}_k(n)=n^k</math>.
			* The [[divisor function]] <math>\sigma_k</math> has <math>(\sigma_k)_p(x)=\frac{1}{1-(1+p^k) x+p^kx^2}.</math>

			==See also==
			* [[Bell numbers]]

			==References==
			* {{Apostol IANT}}

			[[Category:Arithmetic functions]]
			[[Category:Mathematical series]]

en>John of Reading: Typo and General fixing, replaced: concides → coincides using AWB

2012-07-22T17:38:24Z

Typo and General fixing, replaced: concides → coincides using AWB

New page

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Partial isometry - Revision history

en>Freeze S: Preface cleaned up.

en>Mr. Granger: Repairing links to disambiguation pages - You can help!

en>Noix07 at 14:08, 23 January 2014

en>John of Reading: Typo and General fixing, replaced: concides → coincides using AWB