THE JOB DESCRIPTION of a mayor is by no means self-evident; it depends on what problems the city faces and what its government actually can do. In the District, for example, it was of great importance several years ago for a mayor to work to see that blacks were treated fairly by city employees and officials in all departments, to wipe out the legacy of segregation. That task was largely accomplished some time ago. It is less important than it once was also that the mayor be able to get along with conservative southerners in Congress.

Another item not at the top of the next mayor's agenda is education, not because schools are unimportant, but rather because 1) the mayor's authority over the public schools is limited and 2) a newly elected school board and fairly new superintendent are meeting their responsibilities to educate the city's youngsters. Even so, we do hasten to add that a mayor can provide leadership in improving the public schools through mayoral influence on the Board of Education's budget and by the tone and example he (or she) can set because of the visibility of the office.

So what are the areas in which it is reasonable to expect the next mayor to act? Here are our own major items for the mayoral agenda:

Routine government services. The District of Columbia provides most services provided elsewhere by both state and municipal governments, and the fact is that the District's performance in many respects compares favorably with those of other large cities and small states. But it is also a fact that grumbling about the District government's inefficiency and incompetence is about as common around here as grumbling about humidity in August, and not without cause. You may remember that opponents of home rule for the District argued that a government responsible to District residents could not perform routine government services efficiently. We disagreed then and disagree now. But we also think there is much for the mayor elected in 1982 to do to disprove finally this old and disreputable charge and to discredit the false idea that the people of this city can't govern themselves.

Budgeting. Four years ago the District of Columbia's books were unauditable. Since then there has been considerable improvement. But, as the recent dispute over whether there is really a deficit in the District budget shows, the need for rigorous budgeting is continuous. Washington spends as much per capita for local government as any major city and state combined, for a variety of historical reasons, few if any of which are the fault of anyone currently active in District politics. It needs to spend money more efficiently and knowledgeably. To do that, the District needs strong management and rigorous budgeting.

Economic development. In 1982, we learned that Washington is no longer recession-proof. The city's tax base continues to grow largely because of office development that results far more from the growth of government-related businesses than from anything city government has done. But that growth seems to be slowing, and in any case does not seem to provide the kind of entry-level jobs that the high youth unemployment rate suggests is the District's greatest need right now. The next mayor needs to use his or her imagination and to inspire businessmen and investors to create the new jobs Washington needs. The Convention Center, if properly exploited, could help supply many such jobs; but others will be needed as well.

Crime. No one has an easy or simple solution to crime, and no candidate in this election claims otherwise. The response to the recent Post series on the police suggests to us that most District residents feel--as many did not several years ago--that the Metropolitan Police are on their side. Still, whoever is elected mayor will have much to do to ensure that Washington is made safer and that law enforcement works well and works fairly.

Housing. The District government cannot solve all its citizens' housing problems; it simply lacks the resources. But it has to manage public housing projects, stimulate building of new housing and prevent undue hardship to those unable to protect themselves from the workings of the private housing market.

These are all items that we think should be at the top of the agenda of the mayor this city will elect in 1982. You may very well be able to think of more. Those who will be voting in the Democratic primary should, we think, make their choice with some such agenda in mind. We plan to spend the time between now and Sept. 14 offering some judgments and analysis of which candidate might do best these things that should be high on the action list of the mayor of Washington.