How many Sudoku puzzles are there with at least one solution?

Due to Felgenhaur and Jarvis

(see their paper here: http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf)

and Fangying, Mengshi, and Aslasksen

(see their paper here: http://www.math.nus.edu.sg/aslaksen/projects/SHI-ZHANG_Sudoku.pdf)

we know that the total possible number of legal 9x9 Sudoku enumerations is $N_{\text{enumerable}}=6670903752021072936960$.

I will provide an upper bound to the number of puzzles with at least one solution solution, $N_{\text{solvable}}$.

Summing up the number of ways we can replace $n$ squares on a given enumerated grid with $0$'s, and then multiplying by the number of legal grids gives

$$N_{\text{solvable}}<N_{\text{enumerable}}\cdot\sum_{n=0}^{81}\binom{81}{n}=6670903752021072936960\cdot2^{81}=16129255571964761146444296776129424146741329920\approx1.62\times10^{46}$$

This is only an upper bound because it will certainly be an over-count (this is easy to see), so I welcome any refinements to this upper bound as edits below my post.