Justice John Marshall Harlan articulated his "reasonable expectation of privacy test" in his concurring opinion to the Court's 1967 decision in Katz v. United States. The case involved wiretapping, a practice that had not yet been included in the definition of searches under the Fourth Amendment. The question was whether or not authorities needed a warrant to tap a person's phone, and the Court ruled that they did. In concurring with the majority opinion, Harlan wrote that when a person had demonstrated an "expectation of privacy" that society would view as "reasonable," then the state needed a warrant to violate this expectation. So Harlan's test essentially had two parts, one related to individual expectations and the other to societal and legal norms. The majority decision of Court had defined what constituted a search while not completely rejecting warrantless wiretaps as illegal. Harlan's test has been applied to other cases as well, sometimes protecting against searches and other times, like when a person is in an open area, denying this protection.

I assume that you are talking about the test that Harlan proposed in Katz v. United States for when a person has the right to privacy.

If so, Harlan proposed a two-pronged test.  First, he said that the person has to act as if they expect that they are in private.  In other words, if someone puts something out right in front of their window where everyone can see it, they are not acting as if they are in private.  Second, he said that this expectation has to be one that society is willing to recognize as valid.  For example, a person having a conversation in the middle of a crowd could not reasonably expect that the conversation would be private.

This was Harlan's privacy test, which was later adopted by the whole Court.

What is Justice Harlan's formulation of the privacy test?

I assume that you are asking about Justice Harlan’s formulation of the privacy test in his concurring opinion in the case of Katz v. United States.  This was a case about the legality of a wiretap on a telephone booth.  It was decided in 1967.

Harlan’s test has two prongs.  Both of these are subjective.  Taken together, they help us to determine when a search warrant is needed because a person is in a place that is private.  The first prong of Harlan’s test has to do with the state of mind of the person whose actions are being observed.  In order to have privacy, the person must act in a way that shows that they expect privacy.  The second prong has to do with societal attitudes.  The person must be in a place where society in general would typically think that the person should have privacy.  It is not enough for the person to think they have privacy.  The situation must also be one in which most people (not just the person in question) would expect to have privacy.

Thus, Harlan’s test in Katz is formulated with reference to the belief of the person whose actions are being observed and the beliefs of society as a whole.

What is Justice Harlan's formulation of the privacy test?

In Katz v. US, Justice John Harlan wrote in his concurring opinion that authorities would have to get a warrant to monitor communication from any place where a person might have a "reasonable expectation of privacy." He was agreeing with, and slightly expanding, the majority ruling that a telephone booth was one such place, because the person in question had made an effort (shutting the door behind him) to ensure his privacy. This essentially meant that privacy was not simply something one could only enjoy in one's own home. Harlan reinforced the court's decision that electronic as well as physical violations of privacy required a warrant. He further ruled that "warrants are the general rule, to which the legitimate needs of law enforcement may demand specific exceptions," suggesting that while there would be many exceptions, the onus would be on law enforcement to demonstrate them if the Fourth Amendment protection against unreasonable searches and seizures was to have any real force. So Harlan strengthened and expanded the definition of privacy through the "reasonable expectation" test.