Sigmafunktionen är inom talteorin en aritmetisk funktion som definieras som summan av
m
{\displaystyle m}
delare till ett positivt heltal
n
{\textstyle n}
σ
m
(
n
)
=
∑
d
|
n
d
m
{\displaystyle \sigma _{m}(n)=\sum _{d|n}d^{m}}
Sigmafunktionen är multiplikativ (men inte komplett multiplikativ ) och kan därmed beräknas utifrån primfaktoriseringen av
n
{\textstyle n}
σ
m
(
p
1
a
1
.
.
.
p
r
a
r
)
=
∏
i
=
1
r
p
i
m
(
a
i
+
1
)
−
1
p
i
m
−
1
{\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}}}
Dirichletserier innehållande sigmafunktionen är
∑
n
=
1
∞
σ
a
(
n
)
n
s
=
ζ
(
s
)
ζ
(
s
−
a
)
,
{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a),}
som för
a
=
0
{\textstyle a=0}
∑
n
=
1
∞
d
(
n
)
n
s
=
ζ
2
(
s
)
,
{\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s),}
och
∑
n
=
1
∞
σ
a
(
n
2
)
n
s
=
ζ
(
s
)
ζ
(
s
−
a
)
ζ
(
s
−
2
a
)
ζ
(
2
s
−
2
a
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n^{2})}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-2a)}{\zeta (2s-2a)}}}
∑
n
=
1
∞
σ
a
(
n
)
σ
b
(
n
)
n
s
=
ζ
(
s
)
ζ
(
s
−
a
)
ζ
(
s
−
b
)
ζ
(
s
−
a
−
b
)
ζ
(
2
s
−
a
−
b
)
.
{\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
∑
n
=
1
∞
q
n
σ
a
(
n
)
=
∑
n
=
1
∞
n
a
q
n
1
−
q
n
.
{\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}.}
σ
3
(
n
)
=
1
5
{
6
n
σ
1
(
n
)
−
σ
1
(
n
)
+
12
∑
0
<
k
<
n
σ
1
(
k
)
σ
1
(
n
−
k
)
}
.
{\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\}.\;}
σ
5
(
n
)
=
1
21
{
10
(
3
n
−
1
)
σ
3
(
n
)
+
σ
1
(
n
)
+
240
∑
0
<
k
<
n
σ
1
(
k
)
σ
3
(
n
−
k
)
}
.
{\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\}.\;}
σ
7
(
n
)
=
1
20
{
21
(
2
n
−
1
)
σ
5
(
n
)
−
σ
1
(
n
)
+
504
∑
0
<
k
<
n
σ
1
(
k
)
σ
5
(
n
−
k
)
}
=
σ
3
(
n
)
+
120
∑
0
<
k
<
n
σ
3
(
k
)
σ
3
(
n
−
k
)
.
{\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}}}
σ
9
(
n
)
=
1
11
{
10
(
3
n
−
2
)
σ
7
(
n
)
+
σ
1
(
n
)
+
480
∑
0
<
k
<
n
σ
1
(
k
)
σ
7
(
n
−
k
)
}
=
1
11
{
21
σ
5
(
n
)
−
10
σ
3
(
n
)
+
5040
∑
0
<
k
<
n
σ
3
(
k
)
σ
5
(
n
−
k
)
}
.
{\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}}}
τ
(
n
)
=
65
756
σ
11
(
n
)
+
691
756
σ
5
(
n
)
−
691
3
∑
0
<
k
<
n
σ
5
(
k
)
σ
5
(
n
−
k
)
,
{\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),\;}
τ
(
n
)
{\textstyle \tau (n)}
Ramanujans taufunktion .
∑
δ
|
n
d
3
(
δ
)
=
(
∑
δ
|
n
d
(
δ
)
)
2
{\displaystyle \sum _{\delta |n}d^{\;3}(\delta )=\left(\sum _{\delta |n}d(\delta )\right)^{2}\;}
d
(
u
v
)
=
∑
δ
|
gcd
(
u
,
v
)
μ
(
δ
)
d
(
u
δ
)
d
(
v
δ
)
{\displaystyle d(uv)=\sum _{\delta \;|\gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right)\;}
σ
k
(
u
)
σ
k
(
v
)
=
∑
δ
|
gcd
(
u
,
v
)
δ
k
σ
k
(
u
v
δ
2
)
{\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \;|\gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right)\;}
Akbary, Amir; Friggstad, Zachary (2009), ”Superabundant numbers and the Riemann hypothesis” , American Mathematical Monthly 116 (3): 273–275, doi :10.4169/193009709X470128 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 ISBN 0-471-80634-X Lagarias, Jeffrey C. (2002), ”An elementary problem equivalent to the Riemann hypothesis”, The American Mathematical Monthly doi :10.2307/2695443 ISSN 0002-9890 Long, Calvin T. (1972), Elementary Introduction to Number Theory 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 Prentice Hall Robin, Guy (1984), ”Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann”, Journal de Mathématiques Pures et Appliquées 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.
