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From: TCL on 30 Jun 2010 10:53
The derivative of an arithmetic function f:Z^+ -> C is defined as

f'(n)=f(n)log n

Does anyone know who first defined this definition? Dirichlet?
I can't find this info from Google.
From: Gerry Myerson on 30 Jun 2010 19:31
In article
<271b2a4b-1624-4f50-a1d8-2c48b39cfef1(a)y4g2000yqy.googlegroups.com>,
TCL <tlim1(a)cox.net> wrote:

> The derivative of an arithmetic function f:Z^+ -> C is defined as
>
> f'(n)=f(n)log n
>
> Does anyone know who first defined this definition? Dirichlet?
> I can't find this info from Google.

Pentti Haukkanen, of Tampere, has written on this (although I think
he uses f'(n) = - f(n) log n), maybe you should try to contact him.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Han de Bruijn on 1 Jul 2010 02:59
On Jul 1, 1:31 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <271b2a4b-1624-4f50-a1d8-2c48b39cf...(a)y4g2000yqy.googlegroups.com>,
>
>  TCL <tl...(a)cox.net> wrote:
> > The derivative of an arithmetic function f:Z^+ -> C is defined as
>
> > f'(n)=f(n)log n
>
> > Does anyone know who first defined this definition? Dirichlet?
> > I can't find this info from Google.
>
> Pentti Haukkanen, of Tampere, has written on this (although I think
> he uses f'(n) = - f(n) log n), maybe you should try to contact him.
>
> --
> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)

Any idea about the heuristics or where it's good for?

Han de Bruijn
From: David C. Ullrich on 1 Jul 2010 08:44
On Thu, 01 Jul 2010 09:31:05 +1000, Gerry Myerson
<gerry(a)maths.mq.edi.ai.i2u4email> wrote:

>In article
><271b2a4b-1624-4f50-a1d8-2c48b39cfef1(a)y4g2000yqy.googlegroups.com>,
> TCL <tlim1(a)cox.net> wrote:
>
>> The derivative of an arithmetic function f:Z^+ -> C is defined as
>>
>> f'(n)=f(n)log n
>>
>> Does anyone know who first defined this definition? Dirichlet?
>> I can't find this info from Google.
>
>Pentti Haukkanen, of Tampere, has written on this (although I think
>he uses f'(n) = - f(n) log n), maybe you should try to contact him.

Why would that be called a "derivative"?


From: Frederick Williams on 1 Jul 2010 09:14
"David C. Ullrich" wrote:
>
> On Thu, 01 Jul 2010 09:31:05 +1000, Gerry Myerson
> <gerry(a)maths.mq.edi.ai.i2u4email> wrote:
>
> >In article
> ><271b2a4b-1624-4f50-a1d8-2c48b39cfef1(a)y4g2000yqy.googlegroups.com>,
> > TCL <tlim1(a)cox.net> wrote:
> >
> >> The derivative of an arithmetic function f:Z^+ -> C is defined as
> >>
> >> f'(n)=f(n)log n
> >>
> >> Does anyone know who first defined this definition? Dirichlet?
> >> I can't find this info from Google.
> >
> >Pentti Haukkanen, of Tampere, has written on this (although I think
> >he uses f'(n) = - f(n) log n), maybe you should try to contact him.
>
> Why would that be called a "derivative"?

A quote from Tom Apostol's _Introduction to Analytic Number Theory_,
section 2.18:

This concept of derivative shares many of the properties of the
ordinary derivative discussed in elementary calculus. For example,
the usual rules for differentiating sums and products also hold if
the products are Dirichlet products.

Theorem 2.26 If f and g are arithmetical functions we have:
(a) (f + g)' = f' + g'.
(b) (f*g)' = f'*g + f*g'.
(c) (f^{-1})' = -f'*(f*f)^{-1}, provided that f(1) = 0.

[* is the Dirichlet product, natrulich.]

--
I can't go on, I'll go on.
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