Sigmafunktionen

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

σ_{m} (n) = \sum_{d | n} d^{m}

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

σ_{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

\sum_{n = 1}^{\infty} \frac{σ_{a} (n)}{n^{s}} = ζ (s) ζ (s - a),

som för $a = 0$ blir

\sum_{n = 1}^{\infty} \frac{d (n)}{n^{s}} = ζ^{2} (s),

och

\sum_{n = 1}^{\infty} \frac{σ_{a} (n^{2})}{n^{s}} = \frac{ζ (s) ζ (s - a) ζ (s - 2 a)}{ζ (2 s - 2 a)}

\sum_{n = 1}^{\infty} \frac{σ_{a} (n) σ_{b} (n)}{n^{s}} = \frac{ζ (s) ζ (s - a) ζ (s - b) ζ (s - a - b)}{ζ (2 s - a - b)} .

En Lambertserie är

\sum_{n = 1}^{\infty} q^{n} σ_{a} (n) = \sum_{n = 1}^{\infty} \frac{n^{a} q^{n}}{1 - q^{n}} .

Identiteter för sigmafunktionen

σ_{3} (n) = \frac{1}{5} {6 n σ_{1} (n) - σ_{1} (n) + 12 \sum_{0 < k < n} σ_{1} (k) σ_{1} (n - k)} .

σ_{5} (n) = \frac{1}{21} {10 (3 n - 1) σ_{3} (n) + σ_{1} (n) + 240 \sum_{0 < k < n} σ_{1} (k) σ_{3} (n - k)} .

\begin{matrix} σ_{7} (n) & = \frac{1}{20} {21 (2 n - 1) σ_{5} (n) - σ_{1} (n) + 504 \sum_{0 < k < n} σ_{1} (k) σ_{5} (n - k)} \\ = σ_{3} (n) + 120 \sum_{0 < k < n} σ_{3} (k) σ_{3} (n - k) . \end{matrix}

\begin{matrix} σ_{9} (n) & = \frac{1}{11} {10 (3 n - 2) σ_{7} (n) + σ_{1} (n) + 480 \sum_{0 < k < n} σ_{1} (k) σ_{7} (n - k)} \\ = \frac{1}{11} {21 σ_{5} (n) - 10 σ_{3} (n) + 5040 \sum_{0 < k < n} σ_{3} (k) σ_{5} (n - k)} . \end{matrix}

τ (n) = \frac{65}{756} σ_{11} (n) + \frac{691}{756} σ_{5} (n) - \frac{691}{3} \sum_{0 < k < n} σ_{5} (k) σ_{5} (n - k),

där

τ (n)

är Ramanujans taufunktion.

\sum_{δ | n} d^{3} (δ) = {(\sum_{δ | n} d (δ))}^{2}

d (u v) = \sum_{δ | gcd (u, v)} μ (δ) d (\frac{u}{δ}) d (\frac{v}{δ})

σ_{k} (u) σ_{k} (v) = \sum_{δ | gcd (u, v)} δ^{k} σ_{k} (\frac{u v}{δ^{2}})

Se även

Källor

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