**Background**

<a href="https://en.wikipedia.org/wiki/Rational_zeta_series">Rational zeta series</a> are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \setminus \{ 0 \} $ for all $n$. 

Every real number $x$ can be expressed by means of such a series. For instance, we have $$ \sum_{n=2}^{\infty} \zeta(n,2) = \sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) =1. \tag{2}\label{2}  $$ 
Here, $\zeta(\cdot)$ is the Riemann zeta function. Also, we have $$ \sum_{n=2}^{\infty} \frac{(\zeta(n)-1)}{n} = 1 - \gamma, \tag{3}\label{3} $$ where $\gamma$ is the Euler-Mascheroni constant. 

**Zeta values in mathematics**

Geometrical and probabilistic interpretations of zeta values at positive integer arguments can be found in <a href="https://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1?noredirect=1&lq=1" title="Geometric / physical / probabilistic interpretations of Riemann zeta(\$n&gt;1\$)?">this MO question</a>. The appear in <a href="https://mathoverflow.net/a/143236/93724">Tamagawa number</a> calculuations, quantum gauge theories, <a href="https://mathoverflow.net/a/112147/93724">distributions</a> of Cauchy random variables, <a href="https://mathoverflow.net/a/218989/93724">probabilities</a> involving $n$-free numbers, and more. 

Mathematical interpretations of the negative Riemann zeta values can be found in this <a href="https://mathoverflow.net/questions/161872/what-are-some-geometric-physical-probabilistic-interpretations-of-the-rieman" title="What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?">MO question</a>. They appear, for instance, in the orbifold characteristic of genus g Riemann surfaces.  

**Rational zeta series at non-integer arguments**

The paragraphs above provide examples of zeta values that come up in mathematics. In the context of this question, I am not interested in such isolated values, but in the rational zeta series described at the top. More specifically, I am interested in rational zeta series with *non-integer arguments*. What I mean by that, is that $p \in \mathbb{Q} \setminus \mathbb{Z}$ in equation \eqref{1}. 

Examples of such series come up in the study of the Spiral of Theodorus. For instance, the *Theodorus constant* is described in equation (16) of the <a href="http://davidbrink.dk/Theodoros.pdf" title="The spiral of Theodorus and sums of zeta-values at the half-integers">following paper</a> (PDF) by David Brink: 

\begin{align} T &:= \frac{1}{2} + \sum_{k=1}^{\infty} (-1)^{k+1} \left[ \zeta\left(k+ \frac{1}{2} \right) -1 \right] \tag{4} \label{4} \\
&= \sum_{x=1}^{\infty} \frac{1}{(x+1)\sqrt{x}}. \tag{5} \label{5} \end{align}

It is related to the sum of the internal angles of Spiral of Theodorus, depicted below: 

$\hspace{4cm}$[![enter image description here][1]][1]


Another example of a rational zeta series with non-integer arguments is *Hlawka's Schneckenconstante* (described on p. 2 of Brink's paper): 

\begin{align} K &:= \sum_{k=0}^{\infty} (-1)^{k} \frac{\zeta(k+1/2)}{2k+1} \tag{6}\label{6} \\
&= \frac{\pi}{4} + \sum_{k=0}^{\infty} (-1)^{k} \frac{\zeta(k+1/2) -1}{2k+1}. \tag{7}\label{7} \end{align}

**Questions**
 
1. What other rational zeta series at non-integer arguments (see equation \eqref{1} with $p \in \mathbb{Q} \setminus \mathbb{Z}$) come up as mathematical constants? How and where do such series appear? 
 2. And what about the case where $p \in \mathbb{R} \setminus \mathbb{Q} $ ? 

**Note**: I've asked a <a href="https://physics.stackexchange.com/questions/825693/what-rational-zeta-series-with-non-integer-arguments-appear-in-physics-if-any" title="What rational zeta series with non-integer arguments appear in physics - if any?">similar question</a> at Physics SE. 

  [1]: https://i.sstatic.net/IxvS2e9Wm.png