Visa källa för Dedekinds psifunktion

Inom [[talteori]]n är '''Dedekinds psifunktion''' den [[aritmetisk funktion|aritmetiska funktionen]]

:<math> \psi(n) = n \prod_{p|n}\left(1+\frac{1}{p}\right).</math>

Funktionen introducerades av [[Richard Dedekind]].

De första värdena av ψ(''n'') är:

:[[1 (tal)|1]], [[3 (tal)|3]], [[4 (tal)|4]], [[6 (tal)|6]], [[6 (tal)|6]], [[12 (tal)|12]], [[8 (tal)|8]], [[12 (tal)|12]], [[12 (tal)|12]], [[18 (tal)|18]], [[12 (tal)|12]], [[24 (tal)|24]], [[14 (tal)|14]], [[24 (tal)|24]], [[24 (tal)|24]], [[24 (tal)|24]], [[18 (tal)|18]], [[36 (tal)|36]], [[20 (tal)|20]], [[36 (tal)|36]], [[32 (tal)|32]], [[36 (tal)|36]], [[24 (tal)|24]], [[48 (tal)|48]], [[30 (tal)|30]], [[42 (tal)|42]], [[36 (tal)|36]], [[48 (tal)|48]], [[30 (tal)|30]], [[72 (tal)|72]], [[32 (tal)|32]], [[48 (tal)|48]], [[48 (tal)|48]], [[54 (tal)|54]], [[48 (tal)|48]], [[72 (tal)|72]], [[38 (tal)|38]], [[60 (tal)|60]], [[56 (tal)|56]], [[72 (tal)|72]], [[42 (tal)|42]], [[96 (tal)|96]], [[44 (tal)|44]], [[72 (tal)|72]], [[72 (tal)|72]], [[72 (tal)|72]], [[48 (tal)|48]], [[96 (tal)|96]], [[56 (tal)|56]], [[90 (tal)|90]], [[72 (tal)|72]], [[84 (tal)|84]], [[54 (tal)|54]], [[108 (tal)|108]], [[72 (tal)|72]], [[96 (tal)|96]], [[80 (tal)|80]], [[90 (tal)|90]], [[60 (tal)|60]], [[144 (tal)|144]], [[62 (tal)|62]], [[96 (tal)|96]], [[96 (tal)|96]], [[96 (tal)|96]], [[84 (tal)|84]], [[144 (tal)|144]], [[68 (tal)|68]], [[108 (tal)|108]], [[96 (tal)|96]], … {{OEIS|id=A001615}}

ψ(''n'') är större än ''n'' för alla ''n'' större än 1 och är jämn för alla ''n'' större än 2. Om ''n'' är ett [[kvadratfritt tal]] är ψ(''n'') = [[sigmafunktionen|σ(''n'')]].

Genererande funktionen av ψ kan ges med hjälp av [[Riemanns zetafunktion]]:

:<math>\sum \frac{\psi(n)}{n^s} = \frac{\zeta(s) \zeta(s-1)}{\zeta(2s)}.</math>

Detta är en konsekvens av <math>\psi= \mathrm{Id} * |\mu| </math> (se [[Dirichletfaltning]]). 

== Referenser ==
* {{Enwp|url=//en.wikipedia.org/w/index.php?title=Dedekind_psi_function&oldid=571167633|artikel=Dedekind psi function|datum=17 december 2013}}
=== Källor ===
* {{Bokref|författare=[[Goro Shimura]]
|titel=Introduction to the Arithmetic Theory of Automorphic Functions
|utgivare=Princeton
|år=1971}} (sida 25, ekvation (1))
* {{Cite arXiv|
|eprint=1012.4817
|first1=N. A.
|last1=Carella
|title=Squarefree Integers And Extreme Values Of Some Arithmetic Functions
|year=2010
}}
* {{Cite arXiv|
|eprint=1106.4038
|first1 = Richard J.
|last1= Mathar
|title=Survey of Dirichlet series of multiplicative arithmetic functions
|year=2011
}} Avsnitt 3.13.2
* {{OEIS2C|A065958}} – &psi;<sub>2</sub>
* {{OEIS2C|A065959}} – &psi;<sub>3</sub>
* {{OEIS2C|A065960}} – &psi;<sub>4</sub>

== Externa länkar ==
* {{MathWorld |title=Dedekind Function |urlname=DedekindFunction}}

[[Kategori:Multiplikativa funktioner]]