The first values of σ 0 ... σ 4
n
=
σ 0 (n)
σ 1 (n)
σ 2 (n)
σ 3 (n)
σ 4 (n)
1
1
1
1
1
1
1
2
2
2
3
5
9
17th
3
3
2
4th
10
28
82
4th
2 2
3
7th
21st
73
273
5
5
2
6th
26th
126
626
6th
2‧3
4th
12
50
252
1394
7th
7th
2
8th
50
344
2402
8th
2 3
4th
15th
85
585
4369
9
3 2
3
13
91
757
6643
10
2‧5
4th
18th
130
1134
10642
11
11
2
12
122
1332
14642
12
2 2 ‧3
6th
28
210
2044
22386
13
13
2
14th
170
2198
28562
14th
2‧7
4th
24
250
3096
40834
15th
3‧5
4th
24
260
3528
51332
16
2 4
5
31
341
4681
69905
17th
17th
2
18th
290
4914
83522
18th
2‧3 2
6th
39
455
6813
112931
19th
19th
2
20th
362
6860
130322
20th
2 2 ‧5
6th
42
546
9198
170898
21st
3‧7
4th
32
500
9632
196964
22nd
2‧11
4th
36
610
11988
248914
23
23
2
24
530
12168
279842
24
2 3 ‧3
8th
60
850
16380
358258
25th
5 2
3
31
651
15751
391251
26th
2-13
4th
42
850
19782
485554
27
3 3
4th
40
820
20440
538084
28
2 2 ‧7
6th
56
1050
25112
655746
29
29
2
30th
842
24390
707282
30th
2‧3‧5
8th
72
1300
31752
872644
In number theory , the divisor function is the function that assigns the sum of its divisors to a natural number , raised to a certain power. It is usually referred to with the Greek letter
σ
{\ displaystyle \ sigma}
.
definition
For a natural number is defined:
n
{\ displaystyle n}
σ
k
(
n
)
: =
∑
d
|
n
d
k
{\ displaystyle \! \ \ sigma _ {k} (n): = \ sum _ {d | n} d ^ {k}}
.
The sum extends over all positive factors of , including and . For example, is therefore
n
{\ displaystyle n}
1
{\ displaystyle 1}
n
{\ displaystyle n}
σ
2
(
6th
)
=
1
2
+
2
2
+
3
2
+
6th
2
=
50.
{\ displaystyle \ sigma _ {2} (6) = 1 ^ {2} + 2 ^ {2} + 3 ^ {2} + 6 ^ {2} = 50.}
Specializations
properties
Values and average order of magnitude of σ
1
Values and average order of magnitude of σ
2
Values and average order of magnitude of σ
3
σ
k
{\ displaystyle \ sigma _ {k}}
is multiplicative , ie for prime applies: .
n
,
m
{\ displaystyle n, m}
σ
k
(
n
⋅
m
)
=
σ
k
(
n
)
⋅
σ
k
(
m
)
{\ displaystyle \ sigma _ {k} (n \ cdot m) = \ sigma _ {k} (n) \ cdot \ sigma _ {k} (m)}
If the prime factorization then is
n
{\ displaystyle n}
n
=
∏
i
=
1
r
p
i
e
i
{\ displaystyle n = \ prod _ {i = 1} ^ {r} {p_ {i} ^ {e_ {i}}}}
σ
k
(
n
)
=
∏
i
=
1
r
∑
j
=
0
e
i
p
i
j
k
{\ displaystyle \ sigma _ {k} (n) = \ prod _ {i = 1} ^ {r} \ sum _ {j = 0} ^ {e_ {i}} {p_ {i} ^ {jk}} }
,
σ
k
(
n
)
=
∏
i
=
1
r
p
i
k
(
e
i
+
1
)
-
1
p
i
k
-
1
{\ displaystyle \ sigma _ {k} (n) = \ prod _ {i = 1} ^ {r} {\ frac {p_ {i} ^ {k (e_ {i} +1)} - 1} {p_ {i} ^ {k} -1}}}
for , and the following applies: .
k
>
0
{\ displaystyle k> 0}
k
=
0
{\ displaystyle k = 0}
σ
0
(
n
)
=
∏
i
=
1
r
(
e
i
+
1
)
{\ displaystyle \ sigma _ {0} (n) = \ prod _ {i = 1} ^ {r} (e_ {i} +1)}
The average order of magnitude of for is , with the Riemann zeta function .
σ
k
{\ displaystyle \ sigma _ {k}}
k
>
0
{\ displaystyle k> 0}
σ
k
(
n
)
∼
ζ
(
k
+
1
)
n
k
{\ displaystyle \ sigma _ {k} (n) \ sim \ zeta (k + 1) n ^ {k}}
ζ
(
s
)
{\ displaystyle \ zeta (s)}
The average order of magnitude of the divisor number function is . More precisely applies with Euler's constant
d
(
n
)
: =
σ
0
(
n
)
{\ displaystyle d (n): = \ sigma _ {0} (n)}
ln
n
{\ displaystyle \ ln n}
C.
{\ displaystyle C}
∑
x
≤
n
d
(
x
)
=
n
ln
n
+
(
2
C.
-
1
)
n
+
O
(
n
)
{\ displaystyle \ sum _ {x \ leq n} d (x) = n \ ln n + (2C-1) n + O ({\ sqrt {n}})}
.
Series formulas
The following applies in particular :
σ
0
{\ displaystyle \ sigma _ {0}}
∑
i
=
1
n
σ
0
(
i
)
=
∑
i
=
1
n
⌊
n
i
⌋
{\ displaystyle \ sum _ {i = 1} ^ {n} \ sigma _ {0} (i) = \ sum _ {i = 1} ^ {n} \ left \ lfloor {\ frac {n} {i} } \ right \ rfloor}
This can be made clear in which one the right sum as writes: If one now by substituted, the addend of the sum to be precisely 1 increases, the share.
∑
i
=
1
∞
⌊
n
i
⌋
{\ displaystyle \ sum _ {i = 1} ^ {\ infty} \ left \ lfloor {\ frac {n} {i}} \ right \ rfloor}
n
{\ displaystyle n}
n
+
1
{\ displaystyle n + 1}
n
+
1
{\ displaystyle n + 1}
Two Dirichlet series with the divisor function are: (p. 285, theorem 291)
∑
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 (sa)}
For
s
>
1
,
s
>
a
+
1
,
{\ displaystyle s> 1, \; s> a + 1,}
which results especially for d ( n ) = σ 0 ( n ):
∑
n
=
1
∞
d
(
n
)
n
s
=
ζ
2
(
s
)
{\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {d (n)} {n ^ {s}}} = \ zeta ^ {2} (s)}
For
s
>
1
{\ displaystyle s> 1}
and (p. 292, sentence 305)
∑
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 (sa) \ zeta (sb) \ zeta (sab)} {\ zeta (2s-ab)}}.}
A Lambert series with the divisor function is:
∑
n
=
1
∞
σ
a
(
n
)
q
n
=
∑
n
=
1
∞
∑
k
=
1
∞
n
a
q
k
n
=
∑
n
=
1
∞
n
a
q
n
1
-
q
n
{\ displaystyle \ sum _ {n = 1} ^ {\ infty} \ sigma _ {a} (n) q ^ {n} = \ sum _ {n = 1} ^ {\ infty} \ sum _ {k = 1} ^ {\ infty} n ^ {a} q ^ {kn} = \ sum _ {n = 1} ^ {\ infty} n ^ {a} {\ frac {q ^ {n}} {1-q ^ {n}}}}
for any complex | q | ≤ 1 and a .
The divisor function can also be explicitly represented as a series using Ramanujan sums :
k
>
0
{\ displaystyle k> 0}
σ
k
(
n
)
=
ζ
(
k
+
1
)
n
k
∑
m
=
1
∞
c
m
(
n
)
m
k
+
1
.
{\ displaystyle \ sigma _ {k} (n) = \ zeta (k + 1) n ^ {k} \ sum _ {m = 1} ^ {\ infty} {\ frac {c_ {m} (n)} {m ^ {k + 1}}}.}
The calculation of the first values of shows the fluctuation around the "mean value" :
c
m
(
n
)
{\ displaystyle c_ {m} (n)}
ζ
(
k
+
1
)
n
k
{\ displaystyle \ zeta (k + 1) n ^ {k}}
σ
k
(
n
)
=
ζ
(
k
+
1
)
n
k
[
1
+
(
-
1
)
n
2
k
+
1
+
2
cos
2
π
n
3
3
k
+
1
+
2
cos
π
n
2
4th
k
+
1
+
⋯
]
{\ displaystyle \ sigma _ {k} (n) = \ zeta (k + 1) n ^ {k} \ left [1 + {\ frac {(-1) ^ {n}} {2 ^ {k + 1 }}} + {\ frac {2 \ cos {\ frac {2 \ pi n} {3}}} {3 ^ {k + 1}}} + {\ frac {2 \ cos {\ frac {\ pi n } {2}}} {4 ^ {k + 1}}} + \ cdots \ right]}
Identities from the Fourier expansion of Eisenstein series
An essential part of the Fourier expansion of Eisenstein series of weight , straight, are the divider functions . From the relations between the Eisenstein series the values of some convolutions of divisor functions can be derived, for example for all :
k
≥
4th
{\ displaystyle k \ geq 4}
σ
k
-
1
{\ displaystyle \ sigma _ {k-1}}
n
∈
N
{\ displaystyle n \ in \ mathbb {N}}
120
∑
m
=
1
n
-
1
σ
3
(
m
)
σ
3
(
n
-
m
)
=
σ
7th
(
n
)
-
σ
3
(
n
)
,
{\ displaystyle 120 \ sum _ {m = 1} ^ {n-1} \ sigma _ {3} (m) \ sigma _ {3} (nm) = \ sigma _ {7} (n) - \ sigma _ {3} (n),}
5040
∑
m
=
1
n
-
1
σ
3
(
m
)
σ
5
(
n
-
m
)
=
11
σ
9
(
n
)
-
21st
σ
5
(
n
)
+
10
σ
3
(
n
)
.
{\ displaystyle 5040 \ sum _ {m = 1} ^ {n-1} \ sigma _ {3} (m) \ sigma _ {5} (nm) = 11 \ sigma _ {9} (n) -21 \ sigma _ {5} (n) +10 \ sigma _ {3} (n).}
swell
↑ Eric W. Weisstein : Divisor Function . In: MathWorld (English).
↑ E. Krätzel: Number theory . VEB Deutscher Verlag der Wissenschaften, Berlin 1981, p. 134 .
↑ Godfrey Harold Hardy , EM Wright: Introduction to Number Theory . R. Oldenbourg, Munich 1958, p. 285, 292 .
↑ E. Krätzel: Number theory . VEB Deutscher Verlag der Wissenschaften, Berlin 1981, p. 130 .
↑ Tom M. Apostol: Modular Functions and Dirichlet Series in Number Theory . 2nd Edition. Springer-Verlag, 1990, p. 140 .
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