Is it possible to study a specific value of a series using a floor function? This question has two parts.
Imagine you have the following series:
$$\frac{i(e^{{i}^{e}}(-1 + s))\zeta'(s)}{k^{i \pi}(1 + (-1 + s) \sum_{j=0}^{\infty}(-1 + s)^{j}\eta_{j})}$$
Does it make sense to analyze the expression using $j = 0$?
Would taking a floor function of the series allow one to evaluate the series at $j=0$?
Consider the additional example of the value for eta:
$$\eta_{j} = \frac{  (-1)^{j} \lim_{x \to \infty}(\frac{-\log^{1+j}{(x)}}{1+j}) \ + \ \sum_{j=1}^{x} \frac{\log^{j}(j)\Lambda(j)}{j} } {j!}$$
Here, the lower bound for $j$ is $1$, does it then make sense to evaluate the expression for eta at $j=1$?
For evaluating eta at $j=1$, I've computed the following:
$$\lim_{x\to\infty} \frac{1}{2}\log^{2}{x}$$
 A: You seem to be misunderstanding sigma notation.  You cannot evaluate a dummy (bound) variable in an expression.  The expressions you give are a bit convoluted, so here's a simpler example.
Consider the expression $$a = \sum_{j=1}^3 j.$$
In fact $a$ is a constant.  The sigma notation here is just short hand for the sum $a = 1 + 2 + 3 = 6$.  Now how do you set $j = 5$, for example, in this expression?  You can't set $j = 5$ in the expression $\sum_{j=1}^3 j =  1 + 2 + 3$, because $j$ is not a variable in the expression at all; the expression is a constant, $6$.  In fact $j$ is a bound variable in the expression, which means it is only used as a short-hand way of describing the sum. The variable $j$ cannot be set to a specific value because $j$ is a different value for each term in the sum.  It's $1$ in the first term, $2$ in the second term, $3$ in the third.
Expressions like $$j \cdot \sum_{j=1}^3 j,$$ in which a variable $j$ is used both as the index in the sum and also appears outside of the sum, are invalid, because it is overloading the symbol $j$ and using it in two different ways.  You can arrive at invalid conclusions.  For example, let $j = 2$.  We have
\begin{align*}
12 &= 6j \\
&= j \cdot (1 + 2 + 3)\\
 &= j  \cdot \sum_{j=1}^3 j\\
 &= \sum_{j=1}^3 j^2 \\
&= 1^2 + 2^2 + 3^2 \\
&= 14.
\end{align*}
For the same reason, your expression for $\eta_j$ is not valid. It uses $j$ both as the index for the sum and as a variable outside the sum.
