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

• Akbary, Amir; Friggstad, Zachary (2009), ”Superabundant numbers and the Riemann hypothesis”, American Mathematical Monthly 116 (3): 273–275, doi:10.4169/193009709X470128, arkiverad från ursprungsadressen den 2014-04-11, https://web.archive.org/web/20140411041855/http://webdocs.cs.ualberta.ca/~zacharyf/papers/superabundant.pdf.
• Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 234 in section 8.8.
• Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011), ”Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis”, INTEGERS: the Electronic Journal of Combinatorial Number Theory 11: A33, http://www.integers-ejcnt.org/l33/l33.pdf
• Choie, YoungJu; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), ”On Robin's criterion for the Riemann hypothesis”, Journal de théorie des nombres de Bordeaux 19 (2): 357–372, doi:10.5802/jtnb.591, ISSN 1246-7405, http://jtnb.cedram.org/item?id=JTNB_2007__19_2_357_0
• Grönwall, Thomas Hakon (1913), ”Some asymptotic expressions in the theory of numbers”, Transactions of the American Mathematical Society 14: 113–122, doi:10.1090/S0002-9947-1913-1500940-6
• Ivić, Aleksandar (1985), The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York etc.: John Wiley & Sons, s. 385–440, ISBN 0-471-80634-X
• Lagarias, Jeffrey C. (2002), ”An elementary problem equivalent to the Riemann hypothesis”, The American Mathematical Monthly 109 (6): 534–543, doi:10.2307/2695443, ISSN 0002-9890
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd), Lexington: D. C. Heath and Company
• Ramanujan, Srinivasa (1997), ”Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin”, The Ramanujan Journal 1 (2): 119–153, doi:10.1023/A:1009764017495, ISSN 1382-4090
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall
• Robin, Guy (1984), ”Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann”, Journal de Mathématiques Pures et Appliquées, Neuvième Série 63 (2): 187–213, ISSN 0021-7824
• Weisstein, Eric W., "Divisor Function", MathWorld. (engelska)
• Weisstein, Eric W., "Robin's Theorem", MathWorld. (engelska)
• 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.