This, too, contains an elementary error, in the sentence
"However, each term here is less than the original, and
hence this is a zero too." The fact that each term is
multiplied by n to the -epsilon allows no such conclusion,
since the terms have different signs and it is only their
magnitude which is known to be decreased by this factor.
Also, even if each term were actually "less than the
original", not just in magnitude, it would not follow
that the terms would add up to zero, since there is no
apparent reason why the sum couldn't be negative.

If the argument were valid, it could be used to prove
that each zero of the zeta function actually corresponds
to an entire line of zeros, which in turn, since the
zeta function is analytic, would imply that it is zero
everywhere.