There are many races Republicans are trying to win this year. Maine may be the one they are trying to give away.

Two weeks ago, Sen. Bob Packwood of Oregon, chairman of the Republican Senatorial Campaign Committee, held a Washington news conference in which he claimed Republicans had good chances of beating incumbent Democratic senators not only in Montana and Nevada, but in such seemingly long-shot contests as those in Mississippi and Ohio.

"What about Maine?" he was asked.

"No comment," he replied, sending shock waves through this state and producing headlines that the national party was writing off Rep. David F. Emery's challenge to Democratic Sen. George J. Mitchell.

To offset the damage of the Packwood news conference, the Emery campaign leaked a set of tracking-poll numbers by a friendly pollster, showing the 34-year-old challenger within 1 1/2 percentage points of the 49-year-old Mitchell, who was appointed to the Senate in 1980 when former senator Edmund S. Muskie became secretary of state.

But Republicans outside the Emery campaign still whispered that Mitchell was more than 10 points up and seemingly holding firm.

It was one more in a series of incidents in which the GOP has seemingly gone out of its way to smooth the Democrat's path to election. Meanwhile, Mitchell appeals openly and successfully for GOP votes.

He cites Senate Appropriations Committee Chairman Mark O. Hatfield (R-Ore.) to rebut Emery's contention that in supporting cutbacks in President Reagan's defense spending plans, he has jeopardized national security or jobs in Maine.

And when Emery challenges Mitchell's opposition to the balanced-budget amendment, as he did in a debate here Thursday, Mitchell quotes the leader of the Maine Republicans, Sen. William S. Cohen, who also voted against the amendment as "a sham . . . a diversion."

Mitchell has the flexibility to seek Republican votes, because he has solid support from the political organization of Democratic Gov. Joseph E. Brennan, who appointed him to the seat and is on his way to an easy second-term victory. He also has strong backing from organized labor and environmental groups.

Emery, on the other hand, has had only token support from his popular Republican colleague, Rep. Olympia Snowe, the holder of Maine's other House seat, who wanted the Senate nomination herself. And in a state famous for ticket-splitting, it is not uncommon to see cars with leftover Reagan strips and fresh Mitchell stickers.

Emery started from behind two months ago, after squandering an early lead in an expensive computerized mail effort so studded with errors that it rebounded. During the summer, mailings to veterans' and elderly groups, criticizing Mitchell, were denounced as "distortions" in the Maine press, damaging the credibility of a challenger who had virtual immunity from criticism during his eight years in the House. Under pressure from national GOP strategists, who had earlier counted the seat as almost certain, Emery dumped his old campaign staff and hired Cohen's two top assistants to run the rescue effort.

Since Tom Daffron became campaign manager and Bob Tyrer press secretary, the Emery campaign has scored more points against Mitchell than it has given away.

New television ads tried to outflank Mitchell by depicting him as an elitist. Though Mitchell is the son of immigrant factory workers, his Bowdoin education, corporate law practice and career as a U.S. attorney and federal judge have stamped him as a member of the establishment.

Emery, by contrast, went to Worcester Polytechnic and was elected twice to the state legislature and four times to the House before trying for the Senate.

Another ad, criticizing Mitchell's vote against the 1982 tax bill, said that the other three members of the Maine delegation had joined with President Reagan, Sen. Edward M. Kennedy (D-Mass.), House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) and others in a bipartisan effort "to close tax loopholes and expose tax evaders."

Such tactics put Mitchell at least momentarily on the defensive. Within 48 hours after Emery's tax bill ad went on the air, Mitchell slapped together a response ad, saying he had not voted for the "biggest tax increase in history" because he "doesn't take orders from Ronald Reagan, Tip O'Neill, Ted Kennedy . . . or anyone except the people of Maine."

But most of all, Mitchell contrasts what he calls his own "independence" with Emery's "down-the-line" support for Reaganomics.

Emery asserts that because of Reaganomics prospects for recovery are "excellent."

When he said that in the debate here, Mitchell held up that day's edition of The Bangor Daily News, a Republican-leaning newspaper, and read the front page headlines.

"Recovery proves elusive as economic growth slows," one said. With friends like that here and in Washington, Emery needs no enemies.