Sigmafunktionen

Sigmafunktionen är inom talteorin en aritmetisk funktion som definieras som summan av ${\displaystyle m}$:te potensen av alla delare till ett positivt heltal ${\textstyle n}$:

${\displaystyle \sigma _{m}(n)=\sum _{d|n}d^{m}}$

Sigmafunktionen är multiplikativ (men inte komplett multiplikativ) och kan därmed beräknas utifrån primfaktoriseringen av ${\textstyle n}$ som

${\displaystyle \sigma _{m}(p_{1}^{a_{1}}...p_{r}^{a_{r}})=\prod _{i=1}^{r}{\frac {p_{i}^{m(a_{i}+1)}-1}{p_{i}^{m}-1}}}$

Genererande funktioner

Dirichletserier innehållande sigmafunktionen är

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a),}$

som för ${\textstyle a=0}$ blir

${\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s),}$

och

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n^{2})}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-2a)}{\zeta (2s-2a)}}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}}.}$

En Lambertserie är

${\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}.}$

Identiteter för sigmafunktionen

${\displaystyle \sigma _{3}(n)={\frac {1}{5}}\left\{6n\sigma _{1}(n)-\sigma _{1}(n)+12\sum _{0<k<n}\sigma _{1}(k)\sigma _{1}(n-k)\right\}.\;}$

${\displaystyle \sigma _{5}(n)={\frac {1}{21}}\left\{10(3n-1)\sigma _{3}(n)+\sigma _{1}(n)+240\sum _{0<k<n}\sigma _{1}(k)\sigma _{3}(n-k)\right\}.\;}$

${\displaystyle {\begin{aligned}\sigma _{7}(n)&={\frac {1}{20}}\left\{21(2n-1)\sigma _{5}(n)-\sigma _{1}(n)+504\sum _{0<k<n}\sigma _{1}(k)\sigma _{5}(n-k)\right\}\\&=\sigma _{3}(n)+120\sum _{0<k<n}\sigma _{3}(k)\sigma _{3}(n-k).\end{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{9}(n)&={\frac {1}{11}}\left\{10(3n-2)\sigma _{7}(n)+\sigma _{1}(n)+480\sum _{0<k<n}\sigma _{1}(k)\sigma _{7}(n-k)\right\}\\&={\frac {1}{11}}\left\{21\sigma _{5}(n)-10\sigma _{3}(n)+5040\sum _{0<k<n}\sigma _{3}(k)\sigma _{5}(n-k)\right\}.\;\end{aligned}}}$

${\displaystyle \tau (n)={\frac {65}{756}}\sigma _{11}(n)+{\frac {691}{756}}\sigma _{5}(n)-{\frac {691}{3}}\sum _{0<k<n}\sigma _{5}(k)\sigma _{5}(n-k),\;}$     där ${\textstyle \tau (n)}$ är Ramanujans taufunktion.

${\displaystyle \sum _{\delta |n}d^{\;3}(\delta )=\left(\sum _{\delta |n}d(\delta )\right)^{2}\;}$

${\displaystyle d(uv)=\sum _{\delta \;|\gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right)\;}$

${\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \;|\gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right)\;}$

Se även

• Delarantal
• Delarsumma

Källor

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