  In this note, we prove a power-saving remainder term for the function
counting $S_3$-sextic number fields. We also give a prediction on the second
main term.
  We also present numerical data on counting functions for $S_3$-sextic number
fields. Our data indicates that our prediction is likely to be correct, and it
also suggests the existence of additional lower order terms which we have not
yet been able to explain.
