What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?

Question

I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in 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 $\log(|\zeta(\frac{1}{2}+it)|)$)

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

Answer 1

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 $\zeta'$.

Answer 2

The main reason why you cannot find a Dirichlet series approximation (even one that holds "most of the time" and "within a certain error") is because $\zeta'(s)$ does not have an Euler product.