It is generally not advisable to remove roots greater than 2-inches in diameter from mature trees implicated in causing lifted sidewalks. Essentially, root pruning should be considered as the method of last resort because root pruning can lead to structural instability, reduced vigor, and possible mortality.
Options in lieu of root pruning to consider include shaving lifted concrete; slabjacking; asphalt ramps; bridging; rerouting the sidewalk; etc.
Because I am not an expert on this subject, may I recommend the following article by The Bartlett Lab Staff / Bartlett Tree Experts:
Best of luck!