Definition df-sgm 27411

Description: Define the sum of positive divisors function (𝑥 σ 𝑛), which is the sum of the xth powers of the positive integer divisors of n, see definition in [ApostolNT] p. 38. For 𝑥 = 0, (𝑥 σ 𝑛) counts the number of divisors of 𝑛, i.e. (0 σ 𝑛) is the divisor function, see remark in [ApostolNT] p. 38. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
df-sgm	⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))

Distinct variable group:   𝑘,𝑛,𝑝,𝑥

Detailed syntax breakdown of Definition df-sgm

Step	Hyp	Ref	Expression
1		csgm 27405	. 2 class σ
2		vx	. . 3 setvar 𝑥
3		vn	. . 3 setvar 𝑛
4		cc 11179	. . 3 class ℂ
5		cn 12316	. . 3 class ℕ
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1569	. . . . . 6 class 𝑝
8	3	cv 1569	. . . . . 6 class 𝑛
9		cdvds 16402	. . . . . 6 class ∥
10	7, 8, 9	wbr 5103	. . . . 5 wff 𝑝 ∥ 𝑛
11	10, 6, 5	crab 3413	. . . 4 class {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛}
12		vk	. . . . . 6 setvar 𝑘
13	12	cv 1569	. . . . 5 class 𝑘
14	2	cv 1569	. . . . 5 class 𝑥
15		ccxp 26865	. . . . 5 class ↑𝑐
16	13, 14, 15	co 7412	. . . 4 class (𝑘↑𝑐𝑥)
17	11, 16, 12	csu 15833	. . 3 class Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥)
18	2, 3, 4, 5, 17	cmpo 7414	. 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))
19	1, 18	wceq 1570	1 wff σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))

This definition is used by:  sgmval  27451  sgmf  27454