(02/15/2023, 12:32 AM)tommy1729 Wrote: 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...