# Are there conjectured closed forms for factorial rational zeta series?

Context
At the bottom of this page from Wolfram Mathworld on the Riemann zeta function, the following constants are defined in equations (133), (137), and (141), respectively:

\begin{align*} C_{1} &:= \sum_{n=2}^{\infty} \frac{\zeta(n)}{n!} \\ & \approx 1.078188729575818482758265436769832381707219, \\\qquad C_{2} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{n!} \\ & \approx 2.407446554790328514709486656223022725582266, \text{ and} \\ C_{3} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!} \\ &\approx 0.869001991962908998811054805561395688892494.\end{align*} It is also stated that "These sums have no known closed-form expression".

However, it could be the case that there are conjectured closed forms for these constants that correspond to the actual values with a high degree of accuracy.

Question: Are there any of such conjectured closed forms for $$C_{1}$$, $$C_{2}$$, and $$C_{3}$$ ?

• For the first one I conjecture $e^{-31/48-23/(48e)+e/24-5/(24\pi)-7\pi/48}\pi^{23e/48-1/6}\sqrt[24]{-\tan(\pi e)}=1.078188729575818482\dots$ :) Aug 9, 2022 at 23:26
• $$C_1 = \int_0^{ + \infty } {\frac{1}{{e^x - 1}}\left[ {\frac{{I_1 (2\sqrt x )}}{{\sqrt x }} - 1} \right]dx}$$
– Gary
Aug 10, 2022 at 1:10
• @Gary Indeed, such formulas are also stated on the Mathworld page (see eq. 134) Aug 10, 2022 at 12:51
• @MaxMuller Oh, I should have checked it.
– Gary
Aug 10, 2022 at 13:51