Mentioning Professors in math PhD applications
This practice is likely, on average, to help more than it hurts.  It does mark you as somewhat more knowledgeable than the typical applicant, and somewhat more focused.  It also ties your name to Professor Z's.  Should Professor Z want you as a student, and should he or she be able to exert influence, that will give your application a huge likelihood of admission.  

However, if Professor Z is politically unpopular or hated in the department (or by the admissions committee), this may actually hurt your application.  Further, if Professor Z is unimpressed with your file, then you might be rejected even if otherwise you would be accepted.



Mentioning a research area can only help, provided that the department is actually strong in that area.