Gov. Brendan T. Byrne of New Jersey, who won a 10-way race for the Democratic gubernatorial nomination, has challenged teh Republican nominee to a debate on the campaign's hottest issue - the state income tax.

Byrne collected about 30 per cent of the vote in the Democratic primary Tuesday, defeating nine other candidates. The Republican primary winner was state Sen. Raymond H. Bateman, victor in a gour-man contest with 54 per cent of the vote.

At a news conference in his office here late Tuesday night, Byrne challenged Bateman to a debate on Sunday.

A spokesman for Bateman said the GOP nominee was leaving Friday for a five-day vacation but would debate Byrne "the following Sunday, any other Sunday or any other day of the week, night or day.

In Chacago, Democrat Micharl A. Bilandic easily won a special election to succeed the late Richard J. Daley as mayor of Chicago, but the voter turnout was teh lowest in mayoral balloting here in more than 50 years.

Bilandic, 54, an attorney and former City Council member known for his bland demeanor and nose-to-the-grind-stone way with facts an d figures amassed 475,169 votes, or 77 per cent, in balloting Tuesday.

Alderman Dennis Block, 29, The only Republican on the 50-member City Council, received 130,945 votes, or 21.3 per cent.