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When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:

Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ be the $k-$ th derivative of the Riemann zeta function. There exist real numbers $E_{1k}(a)\leq0$ and $E_{2k}(a)\geq1$ such that $\zeta^{(k)}(s)-a\neq0$ for $\{s\in C;\ \sigma\leq E_{1k}(a),\ |t|\geq1\}$ and $\{s\in C;\ \sigma\geq E_{2k}(a)\}$. Let $E'_{1k}(a)$ be a sufficiently small constant for example $E'_{1k}(a)\leq min\{E_{1k}(a),-1\}$ such that $\left|\frac{a}{\zeta^{(k)}(s)}\right|<1$ for $\sigma\leq E'_{1k}(a)$ and $|t|\geq 1.$

Let $\delta=-E'_{1k}(a)$ and let $A$ be a positive large real number and $A < D\leq 2A.$

My question is the following:

Can we have an explicit value for $\frac{\zeta^{(k)}(1-s)}{\zeta(s)}$ or are there some estimations for the term $\frac{\zeta^{(k)}(1-s)}{\zeta(s)}$? Note: for $1+\delta+iA\leq s \leq 1+\delta+iD.$

Many thanks.

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    $\begingroup$ What do you mean by explicit value or estimations? $\endgroup$
    – Peter Humphries
    Commented Jan 27, 2017 at 19:19
  • $\begingroup$ @Peter I mean can we find a formula of the form $$\zeta^{(k)}(1-s)/\zeta(s) = A(s)+O(), $$ with $A(s)$ is the principal term and $O$ is the big Oh. $\endgroup$
    – Khadija Mbarki
    Commented Jan 27, 2017 at 19:51
  • $\begingroup$ For what range of $s$? $\endgroup$
    – Peter Humphries
    Commented Jan 27, 2017 at 19:53
  • $\begingroup$ @PeterHumphries I edited the post! $\endgroup$
    – Khadija Mbarki
    Commented Jan 27, 2017 at 20:13
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    $\begingroup$ I'm not sure what you mean by $1+\delta+iA\leq s \leq 1+\delta+iD$, as complex numbers don't satisfy inequalities. Do you mean $s$ on the vertical line segment connecting those two points? $\endgroup$
    – Stopple
    Commented Jan 27, 2017 at 21:04

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