Talk:Multiplicative function

Somebody wrote:

the function d(n) - the number of positive divisors of n, σ(n) - the sum of all the positive divisors of n, σ_k(n) - the sum of the k-th powers of all the positive divisors of n (where k may be any complex number)

How ks can look like? Any example.? Aren't ks just simple natural numbers here?
XJam [2002.04.16]] 2 Tuesday (0)

Normally, the k is taken to be a natural number, that's correct, but the definition works fine for complex numbers, and yields a multiplicative function. Here's how to work with complex exponents, for instance k=i: 5ⁱ = e^ln(5)i = cos(ln(5)) + i sin(ln(5)) (using Eulers formula in complex analysis). AxelBoldt

Yes Axel that is quite OK. But I really don't see any necessity to extend ks over the set of natural numbers. We just have examples for σ_k(n) where k goes from 0, 1, 2 and so on. In fact special cases with k=0 and k=1 are functions d(n) ≡ σ₀(n) and σ(n) ≡ σ₁(n). It is for example with n=144, d(144)= 15, σ(144)=403. For k=2 and for n=2⁴·3²=144 we have:

σ₂(144) = σ₂(2⁴)σ₂(3²) = (1²+2²+4²+8²+16²)(1²+3²+9²) = 341 · 91 = 31031.

Because of the multiplicativity of σ_k(n) it is easier to calculate product of powers than adding all the squares of all the positive divisors of n. Futher on we have:

σ₃(144) = σ₃(2⁴)σ₃(3²) = (1³+2³+4³+8³+16³)(1³+3³+9³) = 4681 &middot 757 = 3543517,

σ₄(144) = ... = 464378915,

σ₅(144) = ... = 64178802493,

σ₆(144) = ... = 9070067614091,

σ₇(144) = ... = 1294620020196997,

σ₈(144) = ... = 185637589303481315,

σ₉(144) = ... = 26676789058694821933,

σ₁₀(144) = ... = 3837572548050547502651,

σ₁₁(144) = ... = 552334249790915518944277 and all the way to "funny" ∞.

We don't have function σ_5i(n) or do we? k is just counter here for this arithmetical function. And finally according to extension into the complex the value for σ_2i(144) is (2¹⁰ⁱ-1)(3⁶ⁱ-1)/(2²ⁱ-1)(3²ⁱ-1). But is this a mathematical reality in this particular case? What in fact ki-th powers of all the positive divisors of n are doing in arithmetical function? I guess nothing. If there are any mathematical meanings, then we should put these facts in the article, don't you agree. --XJam [2002.04.17]] 3 Wednesday (0)

σ_5i(n) is just as real a multiplicative arithmetical function as σ₁₁(n), σ_π(n) or σ_-2(n) for that matter. Personally, I don't know what they are being used for, if at all. AxelBoldt

Yes, strange. Mathematicians can 'produce' something and they can't use it anywhere. Just what Hardy dreamed about and also worked and lived on about. But perhaps we should somehow consider these cases too. Let us just not forget on negative numbers and on complex ones. There were times when people din't know what to do with them and nowadays we can find them in physical reality. And values of σ_ξ(n) become into the set of ξ and are no more part of the set of natural numbers as they are when ξ is natural number. --XJam [2002.04.18]] 4 Thur's day (0)

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Convolution equations are not easier without an argument by no means. We have to be more carefull to read them - because e (which is defined elsewhere) can mean many things in math. And accidentally e also means more famous base of natural logarithm. So I guess we have to write e(n) to distinguish anyway. If I would write in this way at math exam, I wouldn't finish my mechanical engineering after all. I can recall all those long terms in Taylor's formulas for several variables. But every eyes have their own whitewasher... --XJam [2002.04.18]] 4 Thur's day (1st ed)

Good point, I'll call it ε. The arguments were all in the wrong places: it's not f(n)*g(n), but (f*g)(n). AxelBoldt

Yes - I was a little bit confused with (f*g)(n) too and I believe other Wikipedians also. Axel can you please clear one more another thing. Why is multiplicative function more general term than arithmetic one? Or is this just your opinion? I prefer arithmetical one because it tells me more. (And futhermore I am constantly confuseing in English natural number (N) and integer (Z).) --XJam [2002.04.18]] 4 Thur's day (2nd ed)

Every multiplicative function is arithmetic, but not every arithmetic function is multiplicative. So if you tell the reader that a given function is multiplicative, you give them more useful information, for instance that they need to evaluate the function only on the powers of primes. On the other hand, if you tell them that a given function is arithmetic, you haven't told them much, because pretty much everything is an arithmetic function. AxelBoldt

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Partition function P(n) is as example shows non-multiplicative. Because P(1) = 1, but P(10) = 42 and P(2 · 5)=P(2)P(5)=2 · 7 = 14. Can anyone prove this in another way?
XJam [2002.04.18]] 4 Thur's day (3rd ed)

Yup, the partition function needs to be removed. Maybe it could be given as an example of an arithmetic function on that page? AxelBoldt

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I took out the symbol C^N since N includes zero. AxelBoldt 23:31 Sep 19, 2002 (UTC)

1. Yes, for shure. There's that famous difference that here in Wikipedia zero is in N. It should be C^Z+ here, I guess.

2. What about the record of μ = 1^-1. According to Möbius inversion formula should be μ = ε^-1, but I've seen also μ = 1^-1. With this formula Möbius inversion formula can be prooved.

3. How can we 'calculate' in a similar way Id_k? Oh, this is totaly different pf how we produce formulas from other formulas, right?

4 Another thing. Consider this statement:

ε(n): the function defined by ε(n) = 1 if n = 1 and = 0 if n > 1 -- I have seen also that n ≥ 2. => Oh, silly me - this is the same. If natural number must be greater than 1, then the first one is 2... (Forget this :)) --XJamRastafire 00:02 Sep 20, 2002 (UTC)

(2) μ = 1^-1 is correct, but I took it out: it is a mere reformulation of the Möbius inversion formula, and it uses the notation ^-1 which we haven't introduced yet. ε (and not 1!) is the identity element of the group of multiplicative functions, and Möbius says μ * 1 = ε, which means 1^-1 = μ. μ = ε^-1 is incorrect.

(3) I added the formula for Id_k. AxelBoldt 01:35 Sep 20, 2002 (UTC)

Great job Axel. Can you give some more on producing the relation Id_k = σ_k * μ. Where do we start to see it? So simple formulas but yet so deep, aren't they. Ahah, I didn't think on a different notation. I am rather missed up because we don't want to use arguments of functions in full. So, ^-1 does not mean a reciprocal value as it is common known from division or from matrix calculus? I should study D some more... There are a lot of unsolved problems in this particular field as Janko Bračič wrote in his last article. Who says math is boring... --XJamRastafire 02:00 Sep 20, 2002 (UTC)

I have it. We start from:

σ_k = Id_k * 1 and we write in this 'strange' notation:

Id_k = σ_k * 1^-1 = σ_k * μ.

Right? --XJamRastafire 02:12 Sep 20, 2002 (UTC)

Yup, that's right. The ^-1 is not division, it is the "inverse" with respect to * (which is not multiplication). AxelBoldt

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I just took out this example:

• <math>\sigma<math>^*(n): the unitary divisor function, the sum of all the positive unitary divisors of n,

I don't know what unitary divisors are but will be happy to restore the example if pointed to a definition. AxelBoldt 23:49, 19 Oct 2004 (UTC)