You are taking the laplacian of a sum that does not converge absolutely (EDIT: uniformly) on the range 1/2 <Re(z) < 1. I don't think the trick could work because you are doing 2nd order derivatives on the function and THEN you are adding up the results. First you need to show absolute (EDIT: uniformly) convergence for the series for the given range so that you can change the order of differentiation and then once you find the 1st order derivatives, you need to assure absolute (EDIT: uniform) convergence for the results on the interval so that you can change the order again. The trick you are talking about comes after you have made a total of 4 illegal changes of order. It is very unlikely to work.
The Dirichlet series of Zeta is derived by adding two series, yes. However, you are hoping that differentiating would do the trick? I highly doubt it.
1) Assume that I defined a real function of two real variables a, b.
-zeta is the real function zeta of a.
It is known that:
-F(a,b) converges absolutely for a>1, and diverges for a<1/2.
-The status of convergence of F(a,b) (divergent? conditionally convergent? convergent?) is as yet unknown for 1/2 < a < 1 (do not underestimate how difficult the status is to find in that range, for the infinite product)
-F(a, b) happens to equal the value of everywhere for a>1.
2) Applying the Laplacian to in the range a>1 is a perfectly "legal" operation (since both F(a, b) and zeta(2a) are real and always positive in that range) but happens to yield an infinite series over all primes that absolutely converges for a>1/2. This is what puzzles me. If, as you stated, the function F(a, b) clearly diverged in the range 1/2<a<1, then performing a simple step like taking the Laplacian of its logarithm should not give an infinite series that converges in that range, yet it does.
More importantly though, while the series converges in the range 1/2<a<1, does it still equal in that range? (as it does for a>1). This I already should have tried numerically but I lack proper knowledge/software to do it inside the critical strip. If the results don t match, then obviously the problem is over.
With respect to my statement as to "why" the Laplacian of the log converges for a>1/2 (instead of simply converging for a>1), it is precisely because the operation cancels out the terms in . This is shown by using:
where f(a, b) is real and positive.
(See eq 2.16 in attachment for the identity on the RHS, the proof is straightforward but tedious in latex.)
If you compute , you will find that the only terms left of order in equation (5) come from the second order derivatives in . Fortunately these cancel out because .
The laplacian of Log (Zeta Zeta*) vanishes to 0 when one uses the Dirichlet Eta function (being careful not to interchange terms in the series since it is conditionally convergent), leaving only the laplacian of the real function Zeta(2a), expressed as a series which converges for a> 1/2, and is independant of the imaginary part, of course...a trivial equation, as it should be.
Thanks for you help and energy.