Visa källa för Sigmafunktionen

'''Sigmafunktionen''' är inom [[talteori]]n en [[aritmetisk funktion]] som definieras som summan av <math>m</math>:te potensen av alla [[delare]] till ett positivt [[heltal]] <math display="inline">n</math>:

:<math>\sigma_m(n) = \sum_{d|n}d^m</math>

Sigmafunktionen är [[multiplikativ funktion|multiplikativ]] (men inte [[komplett multiplikativ funktion|komplett multiplikativ]]) och kan därmed beräknas utifrån [[primfaktor]]iseringen av <math display="inline">n</math> som

:<math>\sigma_m(p_1^{a_1}...p_r^{a_r}) = \prod_{i=1}^r \frac{p_i^{m(a_i+1)} - 1}{p_i^m-1}</math>

==Genererande funktioner==

[[Dirichletserie]]r innehållande sigmafunktionen är

:<math>\sum_{n=1}^\infty \frac{\sigma_{a}(n)}{n^s} = \zeta(s) \zeta(s-a),</math>

som för <math display="inline">a=0</math> blir

: <math>\sum_{n=1}^\infty \frac{d(n)}{n^s} = \zeta^2(s),</math>

och

:<math>\sum_{n=1}^\infty \frac{\sigma_a(n^2)}{n^s}= \frac{\zeta(s)\zeta(s-a)\zeta(s-2a)}{\zeta(2s-2a)}  </math>

:<math>\sum_{n=1}^\infty \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s) \zeta(s-a) \zeta(s-b) \zeta(s-a-b)}{\zeta(2s-a-b)}.</math>

En [[Lambertserie]] är

:<math>\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n}.</math>

==Identiteter för sigmafunktionen==

:<math>
\sigma_3(n) = \frac{1}{5}\left\{6n\sigma_1(n)-\sigma_1(n) + 12\sum_{0<k<n}\sigma_1(k)\sigma_1(n-k)\right\}.\;
</math>  

:<math>
\sigma_5(n) = \frac{1}{21}\left\{10(3n-1)\sigma_3(n)+\sigma_1(n) + 240\sum_{0<k<n}\sigma_1(k)\sigma_3(n-k)\right\}.\;
</math>

:<math>
\begin{align}
\sigma_7(n)
&=\frac{1}{20}\left\{21(2n-1)\sigma_5(n)-\sigma_1(n) + 504\sum_{0<k<n}\sigma_1(k)\sigma_5(n-k)\right\}\\
&=\sigma_3(n) + 120\sum_{0<k<n}\sigma_3(k)\sigma_3(n-k).
\end{align}
</math> 

:<math>
\begin{align}
\sigma_9(n)
&= \frac{1}{11}\left\{10(3n-2)\sigma_7(n)+\sigma_1(n) + 480\sum_{0<k<n}\sigma_1(k)\sigma_7(n-k)\right\}\\
&= \frac{1}{11}\left\{21\sigma_5(n)-10\sigma_3(n) + 5040\sum_{0<k<n}\sigma_3(k)\sigma_5(n-k)\right\}.\;
\end{align}
</math> 

:<math>
\tau(n) = \frac{65}{756}\sigma_{11}(n) + \frac{691}{756}\sigma_{5}(n) - \frac{691}{3}\sum_{0<k<n}\sigma_5(k)\sigma_5(n-k),\;
</math> &nbsp; &nbsp; där <math display="inline">\tau (n)</math> är [[Ramanujans taufunktion]].

:<math>
\sum_{\delta|n}d^{\;3}(\delta) = \left(\sum_{\delta|n}d(\delta)\right)^2 \;
</math> 

:<math>d(uv) = \sum_{\delta\;|\gcd(u,v)}\mu(\delta)d\left(\frac{u}{\delta}\right)d\left(\frac{v}{\delta}\right) \;
</math> 

:<math>\sigma_k(u)\sigma_k(v) = \sum_{\delta\;|\gcd(u,v)}\delta^k\sigma_k\left(\frac{uv}{\delta^2}\right) \;
</math>

==Se även==
* [[Delarantal]]
* [[Delarsumma]]

== Källor ==
*{{Citation|doi=10.4169/193009709X470128|first1=Amir|last1=Akbary|first2=Zachary|last2=Friggstad|title=Superabundant numbers and the Riemann hypothesis|url=http://webdocs.cs.ualberta.ca/~zacharyf/Papers/superabundant.pdf|journal=American Mathematical Monthly|volume=116|issue=3|year=2009|pages=273–275|archiveurl=https://web.archive.org/web/20140411041855/http://webdocs.cs.ualberta.ca/~zacharyf/papers/superabundant.pdf|archivedate=2014-04-11}}.
* [[Eric Bach|Bach, Eric]]; [[Jeffrey Shallit|Shallit, Jeffrey]], ''Algorithmic Number Theory'', volume 1, 1996, MIT Press.  {{ISBN|0-262-02405-5}}, see page 234 in section 8.8.
* {{Citation | last1=Caveney | first1=Geoffrey | last2=Nicolas | first2=Jean-Louis | last3=Sondow | first3=Jonathan | title=Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis | url=http://www.integers-ejcnt.org/l33/l33.pdf | year=2011 | journal=INTEGERS: the Electronic Journal of Combinatorial Number Theory | volume=11 | pages=A33}}
*{{Citation | last1=Choie | first1=YoungJu | last2=Lichiardopol | first2=Nicolas | last3=Moree | first3=Pieter | last4=Solé | first4=Patrick | title=On Robin's criterion for the Riemann hypothesis | url=http://jtnb.cedram.org/item?id=JTNB_2007__19_2_357_0 | mr=2394891 |arxiv=math.NT/0604314 | year=2007 | journal=Journal de théorie des nombres de Bordeaux | issn=1246-7405 | volume=19 | issue=2 | pages=357–372 | doi=10.5802/jtnb.591}}
* {{Citation | last1=Grönwall | first1=Thomas Hakon | author1-link=Thomas Hakon Grönwall | title=Some asymptotic expressions in the theory of numbers | year=1913 | journal=Transactions of the American Mathematical Society | volume=14 | pages=113–122 | doi=10.1090/S0002-9947-1913-1500940-6}}
* {{citation | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | pages=385–440 }}
* {{Citation | last1=Lagarias | first1=Jeffrey C. | author1-link=Jeffrey C. Lagarias | title=An elementary problem equivalent to the Riemann hypothesis | doi=10.2307/2695443 | jstor=2695443 | mr=1908008 | year=2002 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=109 | issue=6 | pages=534–543}}
* {{citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77-171950 }}
* {{Citation | last=Ramanujan | first=Srinivasa | author-link=Srinivasa Ramanujan | title=Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin | doi=10.1023/A:1009764017495 | mr=1606180 | year=1997 | journal=The Ramanujan Journal | issn=1382-4090 | volume=1 | issue=2 | pages=119–153}}
* {{citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77-81766 }}
* {{Citation | last1=Robin | first1=Guy | title=Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann | mr=774171 | year=1984 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=63 | issue=2 | pages=187–213}}
* {{mathworld|urlname=DivisorFunction|title=Divisor Function}}
* {{mathworld|urlname=RobinsTheorem|title=Robin's Theorem}}
* [http://mathstat.carleton.ca/~williams/papers/pdf/249.pdf Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions] PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.

{{Delbarhetsklasser}}

[[Kategori:Multiplikativa funktioner]]
[[Kategori:Sigmafunktionen| ]]


[[de:Teilersumme]]
[[hu:Osztóösszeg-függvény]]
[[pl:Funkcja σ]]