01/20/2023, 11:02 PM

Here I present one of my favorite functions , my toy zeta function.

Computing zero's, poles, branches, analytic continuation etc is the main goal.

But things are tricky.

changing order of summation , fubini like ideas and such might not be justified.

Symmetry and/vs natural boundaries ??

Summability methods ??

lets begin.

\s\) is a complex number.

\T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} \)

This is well defined for Re(s) > 1.

It seems T(s) = T(-s) but then again we have it only defined for Re(s)>1 for now.

We try analytic continuation by

\\sum_{n>0} (n^s + n^{-s})^{-1} - n^{-s} = - \sum_{n>0} (n^{3s} + n^{s})^{-1}\)

Then assuming \\sum_{n>0} n^{-s} \) has interpretation as zeta(s) then :

\T(s)=\sum_{n>0} (n^{3s} + n^s)^{-1} + zeta(s)\)

This now is suppose to be analytic for Re(s) >1/3.

We still assume T(s) = T(-s) here.

We can continue this proces by noticing

\g(x) = 1/(x + 1/x) = x/(1+x^2) = x - x^3 + x^5 - x^7 + ... = 1/x - 1/x^3 + 1/x^5 - 1/x^7 + ...\)

and g(1/x) = g(x).

Also

\g(x) - 1/x = - 1/(x^3 + x)\)

\g(x) - 1/x + 1/x^3 = 1/(x^5 + x^3)\)

...

\g(x) - x = - 1/(x^{-3} + x)\)

\g(x) - x + x^3 = 1/(x^{-5} + x^{-3})\)

...

So we have error terms too.

letting x = n^(-s) we can use this to get the idea for a zeta series expansion :

\ T(s) = zeta(s) - zeta(3s) + zeta(5s) - zeta(7s) + ... \)

And the error term is suppose to go to zero.

So we consider the infinite sum and use T(s) = T(-s) :

\ T(s) = zeta(s) - zeta(3s) + zeta(5s) - zeta(7s) + ... = \ T(-s) = zeta(-s) - zeta(-3s) + zeta(-5s) - zeta(-7s) + ... \)

by symmetry.

Notice

\T(s)=\sum_{n>0} (n^{ks} + n^{(k-2)s})^{-1} \)

seems to be well defined and meromorphic for Re(s) > 1/k.

We get the feeling since it appears by the above logic that T(s) is defined for Re(s) > 0 ; the equation T(s) = T(-s) seems justified.

But all of these steps are possibly dubious.

changing order of summation , ignoring radius of taylors ( g(x) is expanded in a taylor without mentioning the radius !? ) etc

Then again it is very similar to what we do with the Riemann zeta function.

Changing the taylor expansion points slowly towards values s with small or negative real part might be more formal and better, but very complicated.

I prefer T(s) = zeta(s) - zeta(3s) + ... truncated at stopping at substraction.

But maybe that zeta expansion is not valid or not so good.

What is remarkable is that this T(s) is close to the zeta function for Re(s) >>1.

And this is probably the most fun/complicated function given in closed form in the closest notation.

Enjoy.

regards

tommy1729

Tom Marcel Raes

Computing zero's, poles, branches, analytic continuation etc is the main goal.

But things are tricky.

changing order of summation , fubini like ideas and such might not be justified.

Symmetry and/vs natural boundaries ??

Summability methods ??

lets begin.

\s\) is a complex number.

\T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} \)

This is well defined for Re(s) > 1.

It seems T(s) = T(-s) but then again we have it only defined for Re(s)>1 for now.

We try analytic continuation by

\\sum_{n>0} (n^s + n^{-s})^{-1} - n^{-s} = - \sum_{n>0} (n^{3s} + n^{s})^{-1}\)

Then assuming \\sum_{n>0} n^{-s} \) has interpretation as zeta(s) then :

\T(s)=\sum_{n>0} (n^{3s} + n^s)^{-1} + zeta(s)\)

This now is suppose to be analytic for Re(s) >1/3.

We still assume T(s) = T(-s) here.

We can continue this proces by noticing

\g(x) = 1/(x + 1/x) = x/(1+x^2) = x - x^3 + x^5 - x^7 + ... = 1/x - 1/x^3 + 1/x^5 - 1/x^7 + ...\)

and g(1/x) = g(x).

Also

\g(x) - 1/x = - 1/(x^3 + x)\)

\g(x) - 1/x + 1/x^3 = 1/(x^5 + x^3)\)

...

\g(x) - x = - 1/(x^{-3} + x)\)

\g(x) - x + x^3 = 1/(x^{-5} + x^{-3})\)

...

So we have error terms too.

letting x = n^(-s) we can use this to get the idea for a zeta series expansion :

\ T(s) = zeta(s) - zeta(3s) + zeta(5s) - zeta(7s) + ... \)

And the error term is suppose to go to zero.

So we consider the infinite sum and use T(s) = T(-s) :

\ T(s) = zeta(s) - zeta(3s) + zeta(5s) - zeta(7s) + ... = \ T(-s) = zeta(-s) - zeta(-3s) + zeta(-5s) - zeta(-7s) + ... \)

by symmetry.

Notice

\T(s)=\sum_{n>0} (n^{ks} + n^{(k-2)s})^{-1} \)

seems to be well defined and meromorphic for Re(s) > 1/k.

We get the feeling since it appears by the above logic that T(s) is defined for Re(s) > 0 ; the equation T(s) = T(-s) seems justified.

But all of these steps are possibly dubious.

changing order of summation , ignoring radius of taylors ( g(x) is expanded in a taylor without mentioning the radius !? ) etc

Then again it is very similar to what we do with the Riemann zeta function.

Changing the taylor expansion points slowly towards values s with small or negative real part might be more formal and better, but very complicated.

I prefer T(s) = zeta(s) - zeta(3s) + ... truncated at stopping at substraction.

But maybe that zeta expansion is not valid or not so good.

What is remarkable is that this T(s) is close to the zeta function for Re(s) >>1.

And this is probably the most fun/complicated function given in closed form in the closest notation.

Enjoy.

regards

tommy1729

Tom Marcel Raes