Great post,but:
Problem 1.
<< MaTex` (*$Failed*)
Problem 2:
Integrate[Floor[Log[b, Floor[Ceiling[x]/x]]], {x, 0, Infinity}, Assumptions -> b > 1] (*Integrate[Floor[Log[Floor[Ceiling[x]/x]]/Log[b]], {x, 0, \[Infinity]}, Assumptions -> b > 0]*)(*Give me input*)
I executed this code on Mathematica 12.2.0.
Please follow the link to install MaTeX and required packages. In general you do not need that to run the code because I only use it to display large formula and no computation depends on it.
This is a very tricky integral with very specific strategy not included in automation. Typically breaking down domain for integration to (b^(k+1),b^k) is quite unusual and probably not used anywhere else
Ok thanks.
Mariusz,
For the first problem MaTeXInstall. Then
MaTeXInstall
Needs["MaTeX`"]
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