What can I do about harassing e-mails with false accusations that are being made by my ex-husband?

You can sue your ex-husband for invasion of privacy for the harassing e-mails.  Invasion of privacy is a serious and unreasonable interference with your right to be left alone.  Your damages (the amount you are seeking to recover in a lawsuit for invasion of privacy) could include mental distress, physical illness, harm to social or business interests.  You can also seek punitive damages which are a substantial amount for a wrongful motive on the part of your ex-husband.  Punitive damages are to punish the wrongdoer for intentional and malicious acts.

You said that your ex-husband is also sending these e-mails to others and that the e-mails have false accusations.  If the false accusations are defamatory, you could sue your ex-husband for defamation.  Defamation is a false statement made with knowledge of its falsity communicated to a third person who recognizes that the statement is defamatory and the statement is injurious to your reputation.  Each repetition of the defamatory statement is actionable in a lawsuit for defamation. 

Libel is written defamation.  Slander is spoken defamation.

Your damages (the amount you are seeking to recover in a lawsuit for defamation) would include mental distress, physical illness, medical expense, loss of friends and associates resulting from the defamation.