Closed form expression or asymptotic expansion for (periodic) generalized harmonic numbers?

Question

In contrast with the series $\sum_{k=1}^n k$ and $\sum_{k=1}^n1$, there does not (as far as I know) exist a pure closed form expression (or a nice asymptotic expansion other than the Euler-Maclaurin expansion?) for the generalized harmonic numbers (and in fact the defining series for the Riemann zeta function) $$H_{s}(n)=\sum_{k=1}^n \frac{1}{k^s}\tag{1}$$ with $s\in\mathbb C$ (in particular $\Re(s)=\sigma\in(0,1)$).

My question is however whether there might exist a closed form expression (or maybe an asymptotic expansion other than the Euler-Maclaurin expansion?) of which I call 'periodic generalized harmonic numbers' $P_{\sigma,t}(n)$ and $Q_{\sigma,t}(n)$ (however an expression for one of them would suffice) in which $$P_{\sigma,t}(n)=\sum_{k=1}^n \frac{\cos(t\ln(k))}{k^\sigma}\tag{2a}$$ and $$Q_{\sigma,t}(n)=\sum_{k=1}^n \frac{\sin(t\ln(k))}{k^\sigma}\tag{2b}$$ with $t\in\mathbb R$ and (in particular) $\sigma\in(0,1)$.

The reason why I'm not immediately interested in the Euler-Maclaurin expansion of the three concerning series is that all these expansions involve a complicated expression (involving Bernoulli numbers and derivatives of the concerning terms) which is in fact used to define the Riemann zeta function in some way. I'm in particular looking for a method to derive information from the Riemann zeta function by using another type of asymptotic expansion or even the closed form of the defining series (1) or preferably the (maybe more nicely behaving) 'splitted versions' (2a) and (2b). Is e.g. Fourier analysis an obvious direction to think of?

See for my motivation for this matter also in 4.1.2 of http://fse.studenttheses.ub.rug.nl/19062/1/bMATH_2019_vanderReijdenIS.pdf.

Answer 1

Since $$\sum\limits_{k=1}^n \dfrac{e^{ib\ln k}}{k^a} = \sum\limits_{k=1}^n \dfrac{k^{ib}}{k^a} = \sum\limits_{k=1}^n\dfrac1{k^{a-ib}} = H_{a-ib}(n),$$ then $$P_{a,b}(n) = \Re H_{a-ib}(n),\quad Q_{a,b}(n) = \Im H_{a-ib}(n).$$

Is known that $$H_z(n) = \zeta(z) - \zeta(z,n+1),$$ where $\zeta(z)$ is the Riemann zeta function and $\zeta(z,k)$ is the Hurwitz zeta function.

This allows to choose the most suitable presentation for the goal function.

For example, can be used the integral presentation in the form of $$\zeta(z,n+1) = \dfrac1{\Gamma(z)}\int\limits_0^\infty \dfrac{t^{z-1}}{1-e^{-t}}\,e^{-(n+1)t}\,dt$$ and similarly for Riemann zeta (case $n=-1$).

The periodicity of Hurwitz zeta can be used in the representation of $$\zeta(z, n+1) = 2(2\pi)^{z-1}\Gamma(1-z)\left(\sin\frac{\pi z}2\sum\limits_{k=1}^{\infty} \cos 2\pi(n+1)k) + \cos\frac{\pi z}2\sum\limits_{k=1}^{\infty} \sin2\pi(n+1)k)\right)$$ (also look there).