Convergent expansions for two common trigonometric sums (Q1353739)

The author derives the following formulas for the trigonometric sum \(S_r(t)= \sum^\infty_{n=1} {e^{int}\over n^r}\) which are valid for \(0\leq t\leq 2\pi\) and \(r>1\). For non-integral \(r>1\), \[ S_r(t)= it^{r-1} \Gamma(1- r) e^{-\pi ir/2}+ \sum^\infty_{k=0} {\zeta(r- k)\over k!} (it)^k, \] for integral \(r>1\), \[ S_r(t)= {(-1)^{r-1} e^{i\pi(1- r)/2}\over(r- 1)!} \Biggl[\log{1\over t}+ {\Gamma'(r)\over\Gamma(r)}+ \gamma+ {i\pi\over 2}\Biggr] r^{r-1}+ \sum^\infty_{k= 0}{\zeta(r- k)\over k!} (it)^k, \] where \(\zeta(r- k)\) is the Riemann zeta function and \(\gamma\) denotes Euler's constant. The proof is based on Mellin's inversion formula. An alternate derivation is sketched, using the Lerch transcendent \(\Phi(z,s,v)= \sum^\infty_{n=0} {z^n\over (v+ n)^s}\).