ATLANTA — As Democrat Joe Biden is sworn in Wednesday as the nation’s 46th president, Donald Trump’s most ardent supporters still believe Biden was not legitimately elected after Trump continues to argue the election was stolen.

There is no evidence of the widespread fraud that Trump and his allies have claimed. Republican and Democratic election officials have certified the election as valid. Courts have rejected lawsuit after lawsuit, and a clear majority of Congress has confirmed the final result despite a riotous mob earlier this month that sought to disrupt the process.

So who has claimed what, precisely? What’s the evidence that the 2020 election was valid and Biden is the duly elected president of the United States?

THE ‘MOST SECURE’ ELECTION IN U.S. HISTORY

After a rocky primary season that played out during the coronavirus pandemic, election officials were determined to ensure voters could safely cast their ballots and ramped up operations to handle a massive influx of absentee ballots. Voting absentee has long been available in the U.S., with some states limiting it to certain voters, and the process has safeguards so any ineligible voter or voter casting multiple ballots is caught and prosecuted.

In many places, election officials added drop boxes for voters concerned about widespread mail delays. Others offered curbside voting and a few states opted to send ballots to all registered voters. Although Trump and his allies claimed these changes were designed to rig the election in favor of Democrats, Trump saw more people vote for him in 2020 than four years earlier and Republicans gained seats in Congress.

One of the changes that drew the most scrutiny was the expansion of absentee voting in Pennsylvania, but that was done prior to the pandemic and authorized in a law passed with bipartisan support through the state’s Republican-controlled Legislature.

State and local election officials have called the November election one of the smoothest in recent memory, with voting spread out across days and even weeks rather than a crush of people at polling places on Election Day. Even Trump’s recently departed attorney general, William Barr, said he saw no evidence of widespread fraud.

And the fact that so many people voted using a paper ballot, which guarantees a record in the event of a dispute, prompted a coalition of government and election security officials, including representatives of the U.S. Department of Homeland Security’s cybersecurity agency, to issue a statement calling 2020 the “ most secure ” election.

WORK REVIEWED, RESULTS CERTIFIED

After voting ended, election officials counted the results and used various measures to verify the totals, such as looking at how many people checked in at each precinct and how many ballots were issued to ensure they add up. For mail ballots, election workers reviewed the voter’s registration to ensure they were eligible to vote and many states matched voter signatures against ones on file.

Any discrepancies were investigated, and results were presented to local election boards and eventually to the state to certify the results as accurate. This happened across the country largely without controversy, except for a few instances in which Republicans raised questions. In these cases, the concerns were dismissed as without merit and the elections certified.

In Georgia, Biden’s win was verified three times: with an initial count, a second conducted by hand and then a recount. The state also conducted an audit of voting machines to ensure the votes recorded electronically matched paper records submitted by voters. A limited review of voter signatures found no evidence of widespread illegal voting.

And ultimately, Republican and Democratic governors signed off on the results, including Trump allies Gov. Brian Kemp of Georgia and Gov. Doug Ducey of Arizona who both certified that Biden won their states.

FLOOD OF LITIGATION, CLAIMS OF FRAUD

Trump and his GOP backers have suffered loss after loss in the legal system, all the way up to the U.S. Supreme Court, as they sought to overturn the will of voters in a few states that Trump lost with unsubstantiated claims of widespread fraud. Their cases have drawn quick dismissals and scathing responses from judges, some of whom were appointed by Republicans or Trump himself:

• U.S. District Judge Timothy Batten in Georgia, an appointee of President George W. Bush, rejected a lawsuit alleging votes were manipulated in favor of Biden, saying “they want this court to substitute its judgment for that of two-and-a-half million Georgia voters who voted for Joe Biden and this I am unwilling to do.”

• U.S. Circuit Judge Stephanos Bibas, a former law professor appointed by Trump, wrote, “voters, not lawyers, choose the president. Ballots, not briefs, decide elections,” as his panel refused to grant a request to stop Pennsylvania from certifying its results.

In about 60 cases filed by Trump’s legal team and Republican allies, Trump notched just one small victory in a fight over the deadline to provide missing proof of identification for certain absentee ballots and mail-in ballots in Pennsylvania.

As his legal bids failed, Trump tried to enlist the help of elected officials in key states. He summoned Michigan lawmakers to the White House in an unsuccessful attempt to set aside the vote tally. He asked Georgia’s governor to convene the state legislature to overturn the results and told Georgia’s top elections official to “ find ” enough votes to sway the state in his favor.

Amid all the claims, it was noteworthy that they began and ended with the presidential contest. There have not been widespread calls to redo the entire election, which also included congressional and state legislative seats.

AFTER MOB, CONGRESS CONFIRMS

The steady drumbeat of false claims about widespread fraud made it to the Capitol as Congress met to confirm Biden’s victory, with more than 100 GOP lawmakers supporting objections.

The meeting to complete the Electoral College tally is normally a routine step. But the process turned deadly as a mob of thousands of Trump supporters stormed the building. After the melee was cleared, lawmakers reconvened and confirmed the results before dawn the following day.

In a strong rebuke Tuesday, Senate Republican leader Mitch McConnell said “ the mob was fed lies ” by Trump and others. “They were provoked by the president and other powerful people, and they tried to use fear and violence to stop a specific proceeding of a branch of the federal government,” McConnell said. He vowed a “safe and successful” inauguration of Biden.

Izaguirre reported from Charleston, West Virginia. AP reporter Alanna Durkin Richer contributed from Boston.

Associated Press coverage of voting rights receives support in part from Carnegie Corporation of New York. The AP is solely responsible for this content.