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Definition df-sgm 24541
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 ∣ 𝑝𝑛} (𝑘𝑐𝑥))
Distinct variable group:   𝑘,𝑛,𝑝,𝑥

Detailed syntax breakdown of Definition df-sgm
StepHypRef Expression
1 csgm 24535 . 2 class σ
2 vx . . 3 setvar 𝑥
3 vn . . 3 setvar 𝑛
4 cc 9786 . . 3 class
5 cn 10863 . . 3 class
6 vp . . . . . . 7 setvar 𝑝
76cv 1473 . . . . . 6 class 𝑝
83cv 1473 . . . . . 6 class 𝑛
9 cdvds 14763 . . . . . 6 class
107, 8, 9wbr 4573 . . . . 5 wff 𝑝𝑛
1110, 6, 5crab 2895 . . . 4 class {𝑝 ∈ ℕ ∣ 𝑝𝑛}
12 vk . . . . . 6 setvar 𝑘
1312cv 1473 . . . . 5 class 𝑘
142cv 1473 . . . . 5 class 𝑥
15 ccxp 24019 . . . . 5 class 𝑐
1613, 14, 15co 6523 . . . 4 class (𝑘𝑐𝑥)
1711, 16, 12csu 14206 . . 3 class Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝑛} (𝑘𝑐𝑥)
182, 3, 4, 5, 17cmpt2 6525 . 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝑛} (𝑘𝑐𝑥))
191, 18wceq 1474 1 wff σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝑛} (𝑘𝑐𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  sgmval  24581  sgmf  24584
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