From what I can see the left hand side is the laplacian of the logarithm term and you have shown that it is never zero. I don't see what that would imply for the logarithm let alone for the zeta function.
Here is what I think. You have shown that the laplacian is never zero. Since it is the sum of two second order partial derivatives we have a few cases.
If , then and is greater in magnitude than the first.
If , then can be any real number as long as it has smaller absolute value than the other partial.
So the partials are either both negative or one is negative and the other is non-negative. I don't know how this would imply that the zeta is non-zero.