# multiplicative function

## Primary tabs

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 Arithmetic functions; related numbers; inversion formulas

### What are the units in the monoid?

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 ?

### Re: What are the units in the monoid?

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.

### Re: What are the units in the monoid?

>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.

### Re: What are the units in the monoid?

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.

### Re: What are the units in the monoid?

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.

### Re: What are the units in the monoid?

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.

### totient

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.

### Re: totient

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

### the tau function

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. :-)