Evaluating Arithmetic Functions In The Complex Plane

[tommy1729]

Hmm, this might be non-sense, but I'll write it anyway. 

The goal will be to construct some functions made up of lambda that allow us to compute \( \lambda(n) \) while only evaulating it on the positive integers. One very useful identity we can use is that
\[ g(n) = \sum_{n=1}^\infty  \frac{\lambda(n)}{n^s}  = \frac{\zeta(2s)}{\zeta(s)}\]
Now, let's consider the two functions  
\[ \sum_{n=0}^\infty  \frac{g(-n)}{z^{n+1}} \qquad \sum_{k=1}^\infty \frac{\lambda(k)}{z-k}\]
These are linked in the following way. We have that

\[\star_1 \quad \sum_{n=0}^\infty  \frac{g(-n)}{z^{n+1}} = \sum_{n=0}^\infty \frac{1}{z^{n+1}} \sum_{k=1}^\infty \lambda(k)k^n = \] 
\[\sum_{k=1}^\infty \frac{\lambda(k)}{z} \sum_{n=0}^\infty   (\frac{k}{z})^n= \]
\[\star_2 \quad \sum_{k=1}^\infty \lambda(k) \frac{1}{z-k} \]
...

Im sorry Caleb but as stated that does not make sense.

First you define 


\[ g(n) = \sum_{n=1}^\infty  \frac{\lambda(n)}{n^s}  = \frac{\zeta(2s)}{\zeta(s)}\]

This is a function of s !!

n is your index of summation.

you even get g(n) = zeta(2s)/zeta(s)

the n magically dissapeared.

the things that follow from it are even worse.

you cannot simple toy with these variables n , s , z etc like you do.

I wasted time looking for a typo or so, but it just does not make sense.

***

Ok so you have a natural boundary at the unit circle.

The most logical way to get out is a reflexion formula to get the interior biholomorphic to the exterior of the unit circle.

We also want it to be almost continu near the unit circle.

So we get

f(z) = f( conjugate 1/z ) 
 
or

f(z)  =  conjugate f ( 1/z )

where conjugate is the complex conjugate ofcourse.


( this is somewhat analogue of automorphic functions (or q-series) such as the 2 functional equations satisfied by the dedekind eta function )

This way the unit circle works a bit like a mirror.
This is the most mathematical and physical and logical idea I can see.

However why would that be interesting ?

Notice the riemann mapping theorem strongly limits the ways a circle maps to itself , or the way a function maps to the circle.

If you want non-analytic ( not even anti-analytic ) reflection formulas you need more motivation ( because it seems random ) and you would loose analyticity outside the unit circle ! Which you actually dont want right ?

Not trying to be rude.


regards

tommy1729