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multiplicative function

Defines: 
multiplicative, completely multiplicative, completely multiplicative function, convolution identity function, convolution inverse
Keywords: 
number theory, arithmetic function
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

11A25 no label found

Comments

Why not mention, in passing, that the monoid of functions under Dirichlet convolution contains a submonoid, in fact a group, namely the functions that take at 1 a value different from 0 ?

This directly goes to monoids. I do not quite understand what to correct in entry "multiplicative function". I guess your message is just mediate reference. Let me know. Best regard.

>This shows that the multiplicative functions
>with convolution form a commutative monoid with
>identity element . relations among the
>multiplicative functions discussed above include

In reference to this. I often see the exclusion of the zero function from the multiplicative functions, so any multiplicative function F that is not identically zero must have F(1) = 1 (easy to show). Moreover, under Dirichlet convolution, a function G has a Dirichlet inverse G^-1 whenever G(1) \not= 0, so that the monoid you mention, given appropriate modifications to the definition of a multiplicative functions, is in fact a group.

Oops, your definition of multiplicative already specifies f(1) = 1. This is equivalent to specifying that a multiplicative function cannot be identically zero. So ignore the previous comment concerning changing the definition of multiplicative.

Okay. Thank you anyway for your effort. We all miss something along the way, don't we. So I believe that a subject is explained enough understandable. Best regard.

Ok, I think I'm not being clear.

Change

> this shows that the multiplicative functions with the convolution form a commutative monoid
into
> his shows that the multiplicative functions with the convolution form a commutative group

And I'll be happy.

Nice article. Maybe this is the object in which we should define "totient". Anyway, I wondered for years why Sylvester coined that strange word in 1882.

An arithmetic function f is called a "totient" if g*f=h for some two completely multiplicative functions g and h, where * is the convolution product.

Maybe the word "totient" is a hybrid of "total" (from "totally multiplicative") and "quotient", since we can indeed write f=h/g.

Yes, dear Sylvester, the Master of coining. For sure. Just mail me your propositions and I'll try to put them in. Best regards.

Overall, I think that this entry is very informative and interesting. One comment that I'd like to make is that I have only seen the function that counts the positive divisors of a positive integer as the tau function. Again, a very good entry. :-)

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