What is known about sums of non-integer zeta values?

Question

There is quite a bit of literature that describes how sums of zeta values at integral arguments can be evaluated, see for instance [1] [2]. On p.7 of [3], however, the author notes that he has not been able to find a closed-form expression for any sum of zeta values at the half-integers.

Questions

1. What is known about sums of zeta values at half-integer arguments?
2. Is anything known about sums of zeta values at rational arguments?

Sources

[1] Computational Strategies for the Riemann Zeta Function -- Borwein, Bradley, Crandall (2000) : link.

[2] Some Series for the Zeta and Related Functions -- Adamchick and Srivastava (1998) : link

[3] The Spiral of Theodorus and Sums of Zeta-values at the Half-integers -- Brink (2012) : link

Answer 1

I hope this Lemma will help you

LEMMA. If $f(x)$ have Taylor series around $0$ in $(-a,a)$, $a\geq 1$. If also the Taylor series of $f(x)$ converges absolutely in $x=1$. Then exists constant $C_f$ depending from $f(x)$ such $$ \sum^{N}_{k=1}f\left(\frac{1}{k}\right)=\int^{N}_{1}f\left(\frac{1}{t}\right)dt+C_f+O\left(N^{-1}\right)\textrm{, }N\rightarrow \infty\tag 1 $$ and $$ C_f=f(0)+f'(0)\gamma+\sum^{\infty}_{s=2}\frac{f^{(s)}(0)}{s!}\left(\zeta(s)-\frac{1}{s-1}\right).\tag 2 $$ The proof is in pages 36-39 of link.

Setting weights to above formula (1), we get for $0<\alpha<1$ $$ \sum^{N}_{k=1}f\left(\frac{1}{k}\right)\frac{1}{k^{\alpha}}-\int^{N}_{1}f\left(\frac{1}{t}\right)\frac{1}{t^{\alpha}}dt=C_f(\alpha)+O\left(N^{a-1}\right)\textrm{, }N\rightarrow \infty\tag 3 $$ and $$ C_f(a)=f(0)\left(\zeta(\alpha)-\frac{1}{\alpha-1}\right)+f'(0)\left(\zeta(1+\alpha)-\alpha^{-1}\right)+ $$ $$ +\sum^{\infty}_{s=2}\frac{f^{(s)}(0)}{s!}\left(\zeta(s+\alpha)-\frac{1}{s+\alpha-1}\right).\tag 4 $$ EXAMPLE. With $\alpha=1/2$, $f(t)=\frac{1}{1+\lambda t}$, $|\lambda|<1$, we get $$ \sum^{N}_{k=1}\frac{\sqrt{k}}{k+\lambda}=2\lambda^{1/2}-2+2\lambda^{1/2}\textrm{arccot}(\lambda^{1/2})-2\lambda^{1/2}\arctan\left(\frac{N^{1/2}}{\lambda^{1/2}}\right)+\sum^{\infty}_{s=2}(-1)^s\lambda^{s}\left(\zeta(s+1/2)-\frac{1}{s-1/2}\right)+O(N^{-1/2}) $$