With control of the Senate hanging in the balance, two runoff contests in Georgia remained too close to call early Wednesday as the advantage seesawed back and forth after election officials had tallied more than 4.2 million votes.

The races feature two Republican candidates — Senator Kelly Loeffler and David Perdue, whose Senate term ended on Sunday — trying to fend off their Democratic challengers, Jon Ossoff and the Rev. Raphael Warnock, in contests that have drawn national attention and unprecedented levels of campaign spending.

Democrats saw some significant signs for optimism. Both Senate candidates were winning a larger share of the vote in county after county than President-elect Joseph R. Biden Jr. won in November, when he narrowly carried the state.

Mr. Warnock was running slightly stronger in his race against Ms. Loeffler than Mr. Ossoff was in his contest with Mr. Perdue. After a large tranche of votes in DeKalb County were reported after 11 p.m., many Democrats began to cautiously cheer the possibility of victory.

“With new votes joining the tally, we are on a strong path,” wrote Stacey Abrams, the Democratic activist and former candidate for Georgia, on Twitter.

About 95 percent of voters in both runoff races said that determining control of the Senate was a “major factor” in their vote, according to A.P. voter surveys, with about three in five calling it “the single most important factor.”

Mr. Biden and President Trump both campaigned in the state on Monday, a sign of the high stakes of the races. If either Republican candidate wins, Senator Mitch McConnell, Republican of Kentucky, will remain the majority leader. But if both Democrats win on Tuesday, the party will gain control of the chamber, with Vice President-elect Kamala Harris serving as the tiebreaking vote.

“This could be the most important vote you will ever cast for the rest of your life,” Mr. Trump said at his rally on Monday. The races are the final elections during his presidency, and Mr. Trump, as usual, loomed large.

A majority of voters, 56 percent, said they disapproved of how Mr. Trump has handled the results of the 2020 presidential election, which he lost but has sought to overturn and undermine. At the same time, voters approved of how Secretary of State Brad Raffensperger of Georgia, whom Mr. Trump has sparred with, has handled the situation.

Mr. Biden narrowly carried Georgia in November, becoming the first Democratic presidential candidate to do so since 1992.

But he did not pull ahead of Mr. Trump in the vote counting in Georgia until days after the election. If the Senate races remain tight on Tuesday, the counting — and a final result — would drag later into the week, as it did in the general election in November.

Mr. Biden’s victory in Georgia has lifted the party’s hopes before the runoffs. But Democrats did not fare as well down ballot in November. Mr. Perdue far outpaced Mr. Ossoff by nearly 90,000 votes in their first matchup, suggesting that Democrats still have to make up ground to win the runoff.

The racial makeup of the final electorate will be crucial in a state where Black voters overwhelmingly support Democrats and white voters back Republicans. According to data compiled by georgiavotes.com, Black voters made up a larger share of early voters for the runoff — nearly 31 percent — than they did in the general election, when it was closer to 28 percent.

Mr. Warnock, who is the pastor of Ebenezer Baptist Church in Atlanta, the spiritual home of the Rev. Dr. Martin Luther King Jr., is seeking to become the first Black Democrat elected to the Senate from the South. He and Mr. Ossoff, a 33-year-old documentary film executive, ran in tandem throughout the runoffs.

Mr. Perdue, the former chief executive of Dollar General, and Ms. Loeffler, who was appointed to the Senate a year ago and is seeking a full term, have cast the race as a necessary check on Democratic power in Washington in 2021, though these efforts have been complicated by Mr. Trump’s continued insistence, without evidence, that he won re-election.

With a narrow lead in his critical Senate race, the Rev. Raphael Warnock promised Georgia voters early Wednesday that he would work on their behalf in the Senate, even though the race had not been officially called in his favor.

“We were told that we couldn’t win this election,” said Mr. Warnock, who was leading the Republican incumbent, Kelly Loeffler, by roughly 35,000 votes with about 97 percent of the ballots already counted. “But tonight, we proved that with hope, hard work and the people by our side, anything is possible.”

Mr. Warnock, the pastor at the storied Ebenezer Baptist Church, declared that he “was going to the Senate” to work for all Georgians, though the race has not been decided.

During brief remarks, he recalled his family’s roots in Georgia, noting that he had grown up in Savannah’s housing projects and graduated from Morehouse College. He spoke of his parents, including his mother, who he said “used to pick somebody else’s cotton” as a teenager.

“But the other day, because this is America, the 82-year-old hands that used to pick somebody else’s cotton went to the polls and picked her youngest son to be a United States Senator,” he said.

If he ultimately wins his race, Mr. Warnock would become the first Black Democrat elected to the Senate from the South.

“May my story be an inspiration to some young person who is trying to grasp and grab hold of the American dream.”

“Georgia,” he added, “I am honored by the faith that you have shown in me.”

Speaking around the same time, Ms. Loeffler thanked her supporters and said: “We’re going to win this election.” Still, she acknowledged that the contest was “a game of inches” and that there was still “work to do.”

Mr. Warnock’s race is occurring alongside another tight Senate contest that has yet to be called. That race, between Republican David Perdue and Democrat Jon Ossoff, as of early Wednesday was a virtual tie.

In a statement early Wednesday, Mr. Ossoff’s campaign manager, Ellen Foster, said that “when all the votes are counted we fully expect that Jon Ossoff will have won this election.”

If Democrats manage to win both races, they will take control of the Senate, clearing the way to more easily enact President-elect Joseph R. Biden Jr.’s agenda.

Vice President Mike Pence told President Trump on Tuesday that he did not believe he had the power to block congressional certification of Joseph R. Biden Jr.’s election victory despite Mr. Trump’s baseless insistence that he did, people briefed on the conversation said.

Mr. Pence’s message, delivered during his weekly lunch with the president, came hours after Mr. Trump increased public pressure on the vice president to do his bidding when Congress convenes Wednesday in a joint session to ratify Mr. Biden’s Electoral College win.

“The Vice President has the power to reject fraudulently chosen electors,” Mr. Trump wrote on Twitter on Tuesday morning, an inaccurate assertion that mischaracterized Mr. Pence’s largely formal and constitutionally prescribed role of presiding over the House and Senate.

Mr. Pence does not have the unilateral power to alter the results sent by the states to Congress.

That did not stop Mr. Trump from trying to exert pressure on Mr. Pence on Twitter early Wednesday morning in a post that the social media company flagged as promoting baseless claims about election fraud.

“If Vice President @Mike_Pence comes through for us,” Mr. Trump tweeted, “we will win the Presidency. Many States want to decertify the mistake they made in certifying incorrect & even fraudulent numbers in a process NOT approved by their State Legislatures (which it must be). Mike can send it back!”

Mr. Trump has been trying for days to press the vice president to use his procedural role in the event as an opportunity to change the outcome of the election. It is also a moment that some of Mr. Pence’s advisers have been bracing themselves for since Mr. Trump lost the election and stepped up his baseless claims of widespread voter fraud. At a rally in Georgia on Monday night, the president openly pressured Mr. Pence for the first time to satisfy his demand that the results be changed to benefit him.

Mr. Pence has spent the past several days in a delicate dance, seeking to convey to the president that he does not have the authority to overturn the results of the election, while also placating the president to avoid a rift that could torpedo any hopes Mr. Pence has of running in 2024 as Mr. Trump’s loyal heir.

Even as he sought to make clear that he does not have the power Mr. Trump seems to think he has, Mr. Pence also indicated to the president that he would keep studying the issue up until the final hours before the joint session of Congress begins at 1 p.m. Wednesday, according to the people briefed on their conversation.

The Democrats now appear favored to prevail in both of their Senate races in Georgia, and therefore are the favorites to take control of the U.S. Senate. The two Republican candidates hold small leads in the vote count, and still have a chance to hold onto those advantages, but most of the remaining vote is in Democratic-leaning areas.

The largest block of remaining ballots is the in-person vote in DeKalb County, a heavily Democratic area that includes part of Atlanta. Over all, the two Democratic candidates are favored to win the remaining vote by around nine percentage points, according to estimates from The Upshot.

Democrats benefited from a strong turnout among Black voters, who are on track to represent a much larger share of the electorate than they did in the general election, based on the turnout by precinct and early voting data.

With 5 percent of the vote left to count, the Rev. Raphael Warnock’s projected lead over Kelly Loeffler (1.9 points) is larger than Jon Ossoff’s projected lead (1.0) over David Perdue. It’s hard to say when either race will be called. The Ossoff-Perdue call might have to wait until late absentee and provisional ballots are counted.

The effort by congressional Republicans to deny the presidential results found an echo in the Pennsylvania legislature on Tuesday, when Republicans voted not to seat a Democratic lawmaker who was elected in November and to remove the lieutenant governor, also a Democrat, as the presiding officer of the State Senate.

On a typically ceremonial day of swearing in members, Pennsylvania’s Senate majority refused to seat Senator Jim Brewster, whose narrow victory was officially certified but is being challenged in court.

In a contentious, chaotic session, Republicans also voted to remove Lt. Gov. John Fetterman as the Senate president and to replace him with the top Republican in the chamber.

The lieutenant governor refused at first to leave the rostrum, and for several minutes both he and the Republican voted into his place tried to recognize motions from the floor. Eventually, Mr. Fetterman stepped away.

“I was escorted out,” Mr. Fetterman said in an interview minutes later. “This was a corruption of the fundamental democratic franchise in our state.” He said Mr. Brewster’s win in November was certified by the secretary of the commonwealth and compared the state Republicans’ actions to President Trump’s efforts to subvert the outcome of his race.

Mr. Brewster, who has represented a region outside Pittsburgh for a decade, defeated Nicole Ziccarelli, a Republican, by 69 votes. She is challenging the results in federal court. At issue are several hundred mail ballots that did not have a handwritten date on their outer envelopes. Ms. Ziccarelli lost a challenge in state court.

Jake Corman, the president pro tem of the State Senate and a Republican, told reporters on Monday that his party believed it had to wait for the outcome of the legal challenge before filling the seat. “Our goal is to get it right, not get it fast,” he said.

But Democrats characterized it as a naked power grab. “This idea of having one party decide who is the real victor is a dangerous precedent we’re seeing played out on the national stage,” Mr. Fetterman said.

Jennifer Kocher, a spokeswoman for the Senate Republicans, accused Democrats of creating chaos. “Today, the order and decorum of the Senate were hijacked by Lt. Gov. John Fetterman and members of the Senate Democrat caucus, who failed to adhere to Senate rules,” she said.

The elections in Georgia won’t just determine the fate of the two Senate seats there and the balance of power on Capitol Hill. It will also reveal the extent to which President Trump has disrupted and damaged his own party.

For the past several weeks, Mr. Trump has instigated and intensified a battle in the Georgia Republican universe as he has sought to overturn his loss in the state and blame Republican leaders there for not helping him.

In response, the state’s Republicans have turned on one another, taking sides for or against the president as he continues in his obstinate — some say unlawful — effort to overturn the election results in Georgia, where he lost by nearly 12,000 votes.

The outcome of these Senate runoffs will show, on one level, how Republican voters have reacted to Mr. Trump’s quest to upend what he has falsely called a “rigged” election.

If Republicans ultimately do not turn out in large numbers, the blame will fall at least partly on the president for his efforts to raise doubts about the fairness of the election process.

The extent to which Mr. Trump is willing to go in that effort became fully apparent on Saturday, when he called Brad Raffensperger, Georgia’s Republican secretary of state and a Republican, urging him to “find” votes and recalculate the results of the state’s presidential contest in his favor, ignoring the official finding, already certified by the governor, that he had lost.

It was the culmination of efforts to overturn the election that began nearly two months ago. At every turn, Mr. Raffensperger and other Georgia election officials have debunked the conspiracy theories about voter fraud pushed by the president and his allies. Nonetheless, Mr. Trump renewed his attacks again Tuesday night, baselessly claiming in a tweet that a “a big ‘voter dump’ against the Republican candidates” was forthcoming.

The short-term effect of Mr. Trump’s pressure campaign will become evident as the votes are counted in the runoffs pitting David Perdue and Kelly Loeffler, the two Republicans fighting to keep their Senate seats, against two Democrats, Jon Ossoff and the Rev. Raphael Warnock.

For Republicans in the state, the concern all along has been that Mr. Trump’s effort to undermine the election process will depress turnout in the runoffs, partly because he has stoked beliefs that the system itself is rigged and cannot be trusted.

Bill Crane, a Georgia political operative and commentator, said the president’s tactics, as well as the work of activists in the state who have claimed the general election was rigged, were tamping down Republican turnout. “Georgia is still conflicted about whether we should vote at all,” Mr. Crane said.

Data from early voting showed that the turnout in the runoff election was depressed in heavily Republican areas of the state, though analysts say that Republicans tend to favor voting on Election Day while Democrats are more likely to cast their ballots early.

President-elect Joseph R. Biden Jr. on Tuesday urged Georgians to vote and expressed continued optimism about unifying the nation, even as some Republicans in Congress push to overturn his election.

In an interview on WVEE-FM, an Atlanta radio station, Mr. Biden made a case for the importance of electing the Democratic candidates, Jon Ossoff and the Rev. Raphael Warnock, in the runoff elections on Tuesday for Georgia’s Senate seats.

“I need their votes in the Senate,” Mr. Biden said.

He said he was “feeling really optimistic about today,” and he made a simple request to Georgia residents: “Vote, vote, vote.”

Mr. Biden also made a pitch for Mr. Ossoff and Mr. Warnock in an interview with WFXE-FM in Columbus, Ga., declaring, “So much is at stake.”

The president-elect spoke a day after traveling to Atlanta for a drive-in rally with the two Democratic candidates. If both candidates win, their party will gain control of the Senate, with Vice President-elect Kamala Harris as a tiebreaking vote.

In the WVEE interview, Mr. Biden said their election would allow for the passage of $2,000 stimulus checks, and he suggested that the two Democrats could help provide support for his administration’s efforts to distribute the Covid-19 vaccine.

Mr. Biden said he envisioned establishing “thousands of federally run and federally supported community vaccination centers of various sizes across the country” in locations like high school gyms and N.F.L. stadiums.

And Mr. Biden, who ran for president with a message of bringing the country together and working with both parties, stuck to that theme despite plans among some Republicans in Congress to object to certifying the Electoral College results on Wednesday.

“There are enough really decent Republicans — you’re seeing them step up now in the United States Senate — who don’t want to be part of this Trump Republican Party,” Mr. Biden said, citing Senator Mitt Romney of Utah as one example. “There’s a whole bunch of them.”

Former President George W. Bush, who became one of the first key Republicans to acknowledge the electoral victory of President-elect Joseph R. Biden Jr. in November, will attend Mr. Biden’s inauguration later this month — even as President Trump and his G.O.P. allies refuse to accept the results.

A spokesman for Mr. Bush, the only other living Republican president, announced on Tuesday night that Mr. Bush and former first lady Laura Bush would travel to Washington for the Jan. 20 ceremony.

“I believe this will be the eighth inauguration they’ve had the privilege of attending — President Trump’s being the most recent — and witnessing the peaceful transfer of power is a hallmark of our democracy that never gets old,” the spokesman, Freddy Ford, said on Twitter.

On the same day that Mr. Bush’s office confirmed that he will attend Mr. Biden’s swearing-in, a spokeswoman for the oldest living former president, Jimmy Carter, said on Tuesday that the Democrat would miss the inaugural.

“President and Mrs. Carter will not travel to Washington for the inauguration but have sent their best wishes to President-elect Biden and Vice President-elect Harris and look forward to a successful administration,” Deanna Congileo, a spokeswoman for the Carter Center, said in an email on Tuesday night.

Traveling to Washington for the ceremony, one that has been significantly scaled back because of the coronavirus pandemic, would likely pose a considerable risk to the 96-year-old Mr. Carter.

In 2019, Mr. Carter underwent surgery for a broken hip after a fall at his home and needed stitches above his brow later that year after another fall.

In 2015, Mr. Carter announced that he was cancer free after undergoing treatment for metastatic melanoma that had spread to his brain.

Four years ago, Mr. Carter, whom the Atlanta Journal-Constitution reported has attended every inauguration since his own in 1977, went to Mr. Trump’s swearing-in ceremony.

It was still unclear whether Mr. Trump would attend Mr. Biden’s inauguration.

ATLANTA — Georgia’s Election Day voters braved a bracing January chill on Tuesday, arriving at polling places to make their choices in two Senate runoff races that are among the most consequential in recent American history.

In liberal-leaning Atlanta, Whitney Leonard, 24, walked out of the West Hunter Street Baptist Church in a precinct in the West End neighborhood. Ms. Leonard said she voted for the Democratic challengers Jon Ossoff and the Rev. Raphael Warnock. But she said she was not beholden to the party.

Ms. Leonard said she felt that President Trump had proved himself immature and erratic, and she believed that Democrats taking control of the Senate was crucial to undoing the damage he had caused.

Before the presidential election in November, Ms. Leonard had never voted. Now, Ms. Leonard, who was previously incarcerated, said she was going to vote whenever the opportunity presented itself. “You don’t know how much of a privilege it is to vote until it’s been taken away from you,” she said.

In Dalton, the northwest city where Mr. Trump held a rally on Monday night, a steady flow of Georgians poured into Dalton State College to vote.

Northwest Georgia is a conservative stronghold, and Republicans knew their task was to overcome strong statewide Democratic turnout in early voting and absentee ballots. By Tuesday, Mr. Trump’s message to Republicans to get out and vote appeared to resonate.

“Turnout is high,” said Lane Lewis, 44, as he waited to enter the precinct. “You can tell because there’s never a line.”

The reliance on Election Day turnout is a risky proposition for Republicans, who must contend with Georgia’s changing population and growing urban areas that increasingly vote Democratic. It was also a forced choice — considering much of the Republican base is echoing Mr. Trump’s concerns about voter fraud in the presidential election — when it comes to absentee voting, and many have expressed similarly unfounded doubts about the Senate races.

Mr. Lewis said he waited until Election Day to vote because he trusted it would be counted then. He also said he believed Ms. Loeffler and Mr. Perdue would benefit from Republicans like him, who hold conservative values but have sometimes been put off by Mr. Trump’s language.

When asked if he thought Mr. Biden won Georgia in November, Mr. Lewis said, “I have doubts.”

Some have described their voting choices as a desire for balance or an aversion to having one party controlling two houses of government.

Joy Phenix, 55, voted for gridlock. “They need a backstop,” she said outside a polling place in the affluent Atlanta suburbs in east Cobb County, where a modest line of voters shuffled through all morning. Ms. Phenix said she voted for Mr. Perdue and Ms. Loeffler, but that if Mr. Trump had been elected, she would have voted for the Democrats.

Jasmine Knapp, a 30-year-old Dalton resident, said she was ready for the flood of campaign texts and television advertisements to end. Ms. Knapp, who declined to say who she supported, described herself as a conservative who voted Republican, but said she had not agreed with how some, like Mr. Trump, had claimed the election in November was rigged.

“You always hear every election cycle that this vote matters more than anything,” she said, “but that feels true this time.”

The local authorities in Washington are cautioning residents to avoid potentially violent agitators who are expected to gather downtown on Wednesday to amplify President Trump’s false claims of voter fraud in the general election in November.

Chief Robert J. Contee III of the Metropolitan Police Department said the police had received information that people intended to show up to the demonstrations armed, a violation of local firearm laws.

Mr. Trump is expected to appear at the rally on Wednesday and has encouraged his supporters to travel to Washington for the event. Some of Mr. Trump’s allies, including the conspiracy theorist and conservative radio host Alex Jones and some associates who recently received a pardon from the president, spoke to hundreds of people who crowded into the city’s Freedom Plaza on Tuesday, one day before Congress begins the formal counting of the Electoral College votes.

A spokeswoman for the Eighty Percent Coalition, which was publicizing the event on Tuesday, did not return requests for comment.

Tensions already began to escalate on Tuesday night, with some of Mr. Trump’s supporters clashing with police near Black Lives Matter Plaza, videos of the confrontation showed. The police used pepper spray to repel some of the demonstrators.

On Monday, Enrique Tarrio, the leader of the Proud Boys, a far-right group that has supported Mr. Trump’s efforts to overturn the election results, was arrested on charges of destruction of property stemming from an episode in downtown Washington last month.

Mr. Tarrio pleaded not guilty and was released by Judge Jonathan Pittman of the Superior Court of the District of Columbia. But he was ordered to stay out of Washington until his hearing this summer and faces arrest if he tries to stay for the pro-Trump protests. A lawyer for Mr. Tarrio did not immediately respond to a request for comment.

“Protest organizations and the groups they represent have shown an alarming affinity for violence. Sadly, they have not been shy about suggesting the need for violence,” Marc Elrich, the executive of Montgomery County, Md., said in a statement warning residents to avoid potential clashes between supporters of Mr. Trump and counterprotesters. “There is talk of disrupting the counting of votes in Congress, which would require extreme actions.”

Mayor Muriel Bowser of Washington on Monday requested support from the Army National Guard for the rallies. About 340 National Guard troops are expected to be present for the rallies, and Customs and Border Protection has also placed tactical teams on standby to help protect federal property.

“We will not allow people to incite violence, intimidate our residents or cause destruction in our city,” Ms. Bowser said at a news conference on Monday. “We’re asking D.C. residents and people who live in the region to avoid confrontations with anybody who’s looking for a fight.”

There were a number of violent clashes last month between supporters of Mr. Trump and counterprotesters in Washington, where four people were stabbed.

Two more Republican senators were making plans on Tuesday to object to electoral votes won by President-elect Joseph R. Biden Jr. on Wednesday when Congress meets to formalize his victory.

Senator Ted Cruz, Republican of Texas, plans to object to the certification of Arizona’s Democratic electors, according to a person familiar with his plans. And Senator Kelly Loeffler, Republican of Georgia, intends to object to the electors from her state, according to a person familiar with her thinking.

Mr. Cruz, a possible 2024 presidential candidate, is among 11 senators who have said in recent days that they will challenge the Electoral College results unless Congress agrees to create an independent commission to audit the results. But his earlier statements had been vague as to whether he would lodge a formal objection himself. His plan to object was first reported by The Washington Post.

His decision to do so, along with Ms. Loeffler’s, ensures that the House and the Senate will formally debate whether to overturn the results in at least three states, prolonging what is normally a brief, ceremonial session and forcing at least three votes on whether to overturn Mr. Biden’s victory.

Senator Josh Hawley, Republican of Missouri, plans to object to Pennsylvania’s electors, and other Republican senators could still join the mix.

House Republicans are preparing to object to the electors from another three states — Michigan, Nevada and Wisconsin — but under law, they can force a debate and vote on their challenges only if a senator agrees to join them. None are expected to be successful.

But by objecting, Mr. Cruz and the other Republicans are ensuring that Mr. Trump will get one final stand in Congress to argue his baseless claims of widespread election fraud. Senate Republican leaders fear it will fracture the party.

The person familiar with Mr. Cruz’s thinking, who requested anonymity, said Mr. Cruz was seeking not to overturn the election but to draw attention to his idea of forming an election audit commission. There is little chance that will happen, and every state has already certified the results after verifying their accuracy.

President Trump ordered a U.S. attorney in southern Georgia to serve simultaneously as the top federal prosecutor in Atlanta, the Justice Department announced on Tuesday, a day after the official previously in the Atlanta post left abruptly after Mr. Trump pressed Georgia elections officials on unfounded claims of widespread voter fraud there.

The acting U.S. attorney, Bobby Christine, announced his role in a news release, saying that he would serve, effective immediately, as U.S. attorney in both the Southern and Northern Districts of Georgia.

“On January 4, 2021, by written order of the President, Bobby was named Acting U.S. attorney for the Northern District of Georgia,” the news release said. Mr. Trump nominated Mr. Christine and his predecessor in Atlanta, Byung Pak, to serve as U.S. attorneys in 2017.

Mr. Pak said in a short email to his office on Monday that he was resigning immediately because of unforeseen circumstances, and his spokesman gave no further explanation.

A spokesman for the Justice Department in Washington did not respond to an email seeking comment on whether the acting attorney general, Jeffrey A. Rosen, had asked Mr. Pak to leave or Mr. Christine to oversee the Atlanta office.

Mr. Pak said in a news release that he had sought to be “thoughtful and consistent, and to provide justice for my fellow citizens in a fair, effective and efficient manner” during his tenure, and he thanked Mr. Trump for the opportunity to serve.

People who have spoken with Mr. Pak said that they had expected his departure and that he had been looking for other work. But Mr. Pak’s decision to leave on the same day he announced his departure took people in the department by surprise.

On his weekend phone call with Brad Raffensperger, the Georgia secretary of state, Mr. Trump complained about the quality of the investigation into claims of voter fraud in the state, especially in Fulton County, which includes Atlanta. “You have your never-Trumper U.S. attorney there,” he said.

While Mr. Trump did not criticize Mr. Pak by name, and he was not the only U.S. attorney in Georgia, the president has lately become focused on the specious claim that poll workers in Fulton County manipulated the vote count.

A federal judge in Atlanta on Tuesday denied a last-minute effort by President Trump to decertify Georgia’s election results, handing the president yet another courtroom loss just one day before Congress is scheduled to bring the presidential race to an official end.

The ruling from the bench by Judge Mark H. Cohen denying the emergency petition brought the number of legal defeats that Mr. Trump and his allies have suffered since Election Day to more than 60. The challenges have spanned eight states and dozens of courts, and have become more desperate as the vote in Congress on Wednesday to formally certify the victory of President-elect Joseph R. Biden Jr. has drawn closer.

In a complaint filed just hours before the start of the new year, Mr. Trump and his lawyers asked Judge Cohen to toss the verified results of Georgia’s presidential race, citing a litany of previously debunked fraud allegations. They claimed that officials in Georgia allowed dead people to vote, as well as unregistered voters, convicted felons still serving their sentences, and people who had registered to vote at post office boxes.

Mr. Trump raised many of these false accusations on Saturday in an hourlong phone call in which he pressured Georgia’s secretary of state, Brad Raffensperger, to help him “find” just enough votes in Georgia to win the election. On Monday, another Georgia state official, Gabriel Sterling, held a news conference rebutting nearly all of Mr. Trump’s false claims.

Judge Cohen denied the president’s emergency request at a brief hearing on Tuesday morning that journalists were blocked from covering remotely. While reporters have covered most of the hearings related to election challenges from Mr. Trump and his allies by either phone or video, Mr. Trump’s lawyers did not consent to allowing public access to the remote livestream of the hearing on Tuesday.