I apologize for posting too many discoveries, but Maple Primes makes a great place to immortalize them!

This is so fascinating I couldn't resist letting everyone know it!

We now have both integral and summation forms for both the MRB constant and its analog!

As found by this Cloud notebook. As shown in this Cloud notebook.

Doing a little computer calculus

here

to the above forms of the MRB constant gives integrals with these engrossingly symmetric kernels.

Then, as you can see here, this follows and holds true.

From

we get the amazing

Prime examples.

The symmetry of the last two integrals is notable!

Documented here.

also,

No Im or Re, must be wrong. Wrong! It is shown here.

Makes such sense, must be wrong. Wrong! It's provable from all the above and it works:

https://www.wolframcloud.com/obj/44a621cd-e723-4e42-89b3-7f5f69413749

Also, we have 

A definite integral for the MRB constant follows

Comparison of that with the related definite integral of its integrated analog:

Using the results in the previous message, we have the following triad of pairs of integrals summed from -complex infinity to +complex infinity that all equal the MRB constant (CMRB).

I found the following ineteresting relationship between MRB constant formulas.

The first one then the second and third ones are by simple complex analysis and proven in the previous messages. The last one is worked in this link.

Combining 2 previously mentioned formulas for the MRB constant (CMRB), we get the following.

See the proof here.

Units in MRB constant formulas can be replaced by x ∈ ℂ and ℝ.

A fresh look at the connection between the MRB constant and its integrated analog

I had a very interesting question about the rate of convergence of the MRB constant integral. I thought this computable rate of convergence was worth mentioning in this post about CMRB. Here is a summary.