[NT] more zeta stuff for the fans

[tommy1729]

On mathstack the following conjecture about a dirichlet series was written.

https://math.stackexchange.com/questions...dot3-s-5-c

Since there are " zeta fans " here ...


Regards

tommy1729

I don't think this is extendable beyond \(\Re(s) > 2\)--and if it were it would be very non-obvious. To ask for zeroes before analytically continuing is a recipe for disaster. The sum DOES NOT converge for \(1 < \Re(s) < 2\)--so to say there are zeroes here; when the sum diverges; without an analytic continuation is... nonsense.

The standard way to analytically continue is useless here too. If I write:

\[
P(x) = \sum_{n\le x} p_n\\
\]

Then the tight bounds are:

\[
P(x) = \frac{x^2}{2} \left(\log(x) + \log \log(x) - 3/2 + o(1)\right)\\
\]

So from here we have:

\[
\zeta_P(s) = -s\int_1^\infty P(x)x^{-s-1}\,ds\\
\]

This integral converges for \(\Re(s) > 2\); but once you account for \(P(x)\)'s asymptotics--there's no obvious way to extend it. The error term \(o(1)\) decays too slowly-- about \(1/log^2 x\) to meaningfully allow for a simple continuation. Unless mick has a magical reflection formula up his sleeve, this is pretty pointless...

But that integral transform is designed not to have analytic continuation.

If P(x) = x^3 this would also not converge , yet its related zeta function does as it can be expressed by riemann zeta functions.



In fact if we write p_n as a taylor series 

p(n) = a_0 + a_1 n + a_2 n^2 + ...

Then we get expressions like

A(s) = zeta(s) + a_1 zeta(s-1) + a_2 zeta(s-2) + a_3 zeta(s-3) + ...

which might have analytic continuation then, since zeta does.


Notice this is not like an euler product or the prime zeta function.



And even if we have an natural boundary , that does not explain why the zero's approach the boundaries.


regards

tommy1729