I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in [Dirichlet polynomial](https://encyclopediaofmath.org/wiki/Dirichlet_polynomial) but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet polynomial approximation yield probably for $\log(|\zeta'(\frac{1}{2}+it)|)$ also dosn't have an approximation in Dirichlet polynomial but I do not have complet confidence for that , Now my question here is: What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomiall if it is exists ?


**Edit** $s$ is a complex variable ,note that $0<t\leq T$ , $T$ is large enough, I have added this detail because I missed it when I posted the question.


**Related question:**
([Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$](https://mathoverflow.net/q/356970/51189))


**Note** The motivation of this question is the mean -value estimate of derivative of Riemann zeta function over non trivial zero of Riemann zeta function



What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?
A Dirichlet polynomial is a function of the form
$\sum\_{1\le n \le X} a\_n n^{-s}$, where $a\_n$ are complex numbers and
$s = \sigma + i t$ with $\sigma$ and $t$ real. It is an analytic
function of $s$.


You ask whether certain functions can be approximated by a Dirichlet
polynomial. The specific functions you mention are of the form
$\log(|f(s)|)$ and $\log(|f(\frac12 + i t)|)$. You don't say what
range of $s$, or of $t$, you want to have an approximation, but let's
assume the ranges are reasonably large because you want to use that
approximation to prove something.


Then the answer is 'no' because the functions you want to approximate are
real-valued, but a Dirichlet polynomial (or any analytic function)
cannot have a very small imaginary part throughout a large region.
That explains why $\log(|f(s)|)$ cannot be approximated (by any analytic
function, so in particular a Dirichlet polynomial). A Dirichlet polynomial
cannot have very small imaginary part at $\frac12 + i t$ for a wide range
of $t$, so that is why $\log(|f(\frac12 + i t)|)$ cannot be approximated
(take the imaginary part of each term in the sum, and recognize it as
a sum of trig functions).


It is irrelevant that the function $f$ happens to be $\zeta'$.


