Rudy Giuliani made accusations of fraud that the Trump team has failed to support in court.

Rudolph W. Giuliani speaking to reporters in Washington on Thursday.
Credit…Erin Schaff/The New York Times

At a rambling news conference on Thursday, Rudolph W. Giuliani, President Trump’s personal lawyer, mixed misleading statements, wild conspiracy theories and outright fabrications as he attempted to suggest that Mr. Trump still had a viable pathway to winning the election.

Over and over again, Mr. Giuliani and other members of the president’s legal team suggested that Mr. Trump had evidence to prove that “massive fraud” had been committed in swing states across the country. But Mr. Giuliani himself had undercut that accusation in one high-profile case, telling the federal judge overseeing a suit in Pennsylvania, “This is not a fraud case.”

Mr. Giuliani, speaking at the Republican National Committee’s headquarters in Washington, claimed that Mr. Trump would prevail in the election if only he could get his day in front of a judge.

“Give us a chance to prove it in court and we will,” he said.

The problem? In many of the instances that Mr. Giuliani mentioned, the Trump campaign has already had its chance in court — and failed.

For example, Mr. Giuliani quoted a Detroit poll worker named Jessy Jacob, who submitted an affidavit in the case Costantino v. City of Detroit. Ms. Jacob claimed that she had witnessed poll workers in Detroit, a heavily Democratic city, encouraging voters to cast their ballots for Joseph R. Biden Jr., and that while she was working at “a satellite location” she was instructed by her superiors not to ask for voters’ identification.

On Friday, however, that case was dismissed by a Michigan circuit judge, Timothy M. Kenny, who said that several of the charges it contained were “rife with speculation and guesswork.” Judge Kenny specifically addressed Ms. Jacob’s accusations, saying they were “generalized” and asserted “behavior with no date, location, frequency or names of employees.”

Mr. Giuliani also cited an affidavit in the case from Melissa Carone, a contractor for Dominion Voting Systems, to claim that trucks intended to transport food instead brought “thousands and thousands of ballots” to Detroit. But Judge Kenny had also rejected that story, saying that Ms. Carone’s description of the events at a vote counting center in Detroit “does not square with any of the other affidavits” and that her “allegations simply are not credible.”

In referring to other cases brought by the Trump campaign, Mr. Giuliani made even vaguer accusations and provided no evidence to support them. He claimed that he had more than 100 affidavits alleging voting improprieties in a federal lawsuit the campaign filed in Michigan, Donald J. Trump for President Inc. v. Benson. But the campaign’s own lawyers voluntarily dismissed that suit just hours before Mr. Giuliani held his news conference.

Mr. Giuliani further said that the Trump campaign had affidavits proving that nearly 700,000 mail-in ballots in Pennsylvania had been tainted. But no such affidavits have been filed in the campaign’s federal lawsuit in Pennsylvania that is seeking to halt the certification of the vote there. Moreover, Mr. Giuliani never mentioned the affidavits when he personally appeared at a hearing in the case on Tuesday afternoon.

Mr. Giuliani also claimed that President-elect Joseph R. Biden Jr. admitted a few days before the election to “have the best voter fraud team in the world.” This was a reference to a deceptively edited video that spread on social media. In his original remarks, Mr. Biden was referring to efforts to protect against voter fraud.

Sidney Powell, another lawyer for Mr. Trump, baselessly warned of “massive influence of communist money through Venezuela, Cuba and likely China and the interference with our elections here in the United States.” She claimed that tabulating software used by Dominion and Smartmatic, two voting machine companies, was “created by Hugo Chavez.”

Ms. Powell cited a partially redacted affidavit from an unnamed former military official in Venezuela that accused Smartmatic of helping to rig that country’s elections. But Smartmatic did not provide technology to any battleground state in this year’s presidential election in the United States, nor has it sold any software or hardware to Dominion, which is a competitor. Electronic voting security experts told The New York Times that the affidavit contained no evidence of a rigged election in the United States.

Software made by Dominion was used to count votes in several swing states, and the company has been a target of misinformation from Mr. Trump’s allies, but there is no evidence that the software improperly changed any vote tallies.

Ms. Powell also wrongly claimed that a British baron named Mark Malloch-Brown was one of the “leaders of the Dominion project” as well as the billionaire George Soros’s “No. 2 person” in the United Kingdom. Mr. Malloch-Brown is Smartmatic’s chairman and sits on the board of Mr. Soros’s Open Society Foundations. Mr. Soros has been the target of numerous conspiracy theories propagated on the right, some of them with anti-Semitic overtones.

The Trump campaign has not yet included allegations about Dominion machines in any of its more than 30 lawsuits where the accusations would have to be proven in front of a judge.

Shortly after Mr. Giuliani’s news conference, which was carried live on Fox News, the Republican strategist Karl Rove appeared on the channel and said that Mr. Giuliani should prove his “strange” accusations in court or “withdraw” them immediately.

A voter filled out a ballot during early voting at Wayne County Community College in Detroit last month.
Credit…David Goldman/Associated Press

President Trump’s approach to challenging the election has been scattershot and contradictory, as his campaign demands that courts stop ballots from being counted in certain places while insisting that a more thorough review is necessary in other places.

Confusing as it may seem, essentially his goal is this: to get judges to invalidate the results in enough counties and states so that President-elect Joseph R. Biden Jr.’s lead disappears.

Would judges ever actually do that?

They have before, though never on the scale that the president and his legal team is attempting. There are numerous examples going back hundreds of years in the United States when courts have been asked to toss out the results of elections on the local, state and federal levels. Losing candidates have prevailed for a variety of reasons: because the court determined that the count was off, or that inconsistent standards were applied in processing ballots, or even that there was voter fraud.

But these cases are the exception. And election law experts said that judges have set the bar extremely high. It’s not enough to claim — or even prove — that irregularities occurred. The irregularities have to be significant enough to change the outcome of the race, which is extraordinarily rare.

“The prevailing view today is that courts should not invalidate election results because of problems unless it is shown that the problems were of such magnitude to negate the validity of which candidate prevailed,” said Edward B. Foley, director of election law at Ohio State University’s Moritz College of Law. This is inherently difficult to do, he added, given how hard it is to provide evidence that disputed ballots were cast in favor of a particular candidate.

Professor Foley, whose book “Ballot Battles” provides a history of disputed elections in the United States, described one example that illustrates how difficult it will be for the president to succeed with his claims. In an election with a margin of victory of 10,000, it would not be enough to show that there were 11,000 invalid votes, he said, “because those invalid votes might have split 50-50, not making a difference to the outcome.” (In Arizona, the closest of the major swing states, Mr. Trump trails Mr. Biden by roughly 10,000 votes.)

Mr. Trump has cited cases where irregularities and fraud have led to new elections. But his most recent examples take isolated incidents of small-scale error or fraud and misleadingly apply them to a national election in which more than 150 million ballots were cast. There was the case in Paterson, N.J., earlier this year, for instance, in which a judge recommended a do-over election for a seat on the City Council after evidence surfaced that mail-in ballots had been tampered with. (Just 240 votes separated the first- and second-place candidates.)

And this week in Clark County, Nev., local officials voted to rerun one race for a county commission seat that had a margin of just 10 votes, which Mr. Trump falsely claimed as a “big victory” even though he lost the county by more than 90,000 votes.

Professor Foley traces the nation’s first major ballot-counting dispute back to Philadelphia, the site of some of the legal wrangling today. In 1781, the results of the election for Pennsylvania’s Supreme Executive Council, the state’s executive branch at the time, were contested after allegations that soldiers had been marched to the polls by their commanders and forced to vote for a particular candidate.

The Pennsylvania Supreme Court issued an opinion that laid out a standard for considering voter fraud as the grounds for overturning an election that still largely applies today: Voters should not suffer for the misdeeds of a few bad actors, the judges said. For fraud or irregularities to render an election invalid, the problems would have to be substantial. The vote was not set aside.

There have been more recent cases in which fraud rendered an election invalid. In one high-profile example, a Florida judge voided Miami’s mayoral election in 1998 and ordered a new vote, citing “a pattern of fraudulent, intentional and criminal conduct” in the casting of absentee ballots.

Mr. Trump has had no such luck in persuading judges or local elections officials. His latest setback came on Tuesday when Republican officials in Michigan’s largest county reversed their earlier votes to delay certification of the election results there. The officials tried to change their stance yet again late Wednesday, but it appeared to be too late: All counties in the state have now certified their results, showing Mr. Biden ahead by nearly 150,000 votes.

President Trump also reached out personally to at least one election official in Wayne County, Mich., home of Detroit, who tried to decertify the results there.
Credit…Anna Moneymaker for The New York Times

President Trump on Thursday accelerated his efforts to interfere in the nation’s electoral process, taking the extraordinary step of reaching out directly to Republican state legislators from Michigan and inviting them to the White House on Friday for discussions as the state prepares to certify President-elect Joseph R. Biden Jr. the winner there.

Mr. Trump’s outreach to Republican officials in Michigan represented a remarkable intrusion into state and local politics: a sitting president personally contacting officials who usually play a small and invisible role in a routine process.

The president requested the White House meeting with Mike Shirkey, the State Senate majority leader, and Lee Chatfield, the speaker of the Michigan House, and they will sit down with him on Friday afternoon, according to a person briefed on the arrangements. It is not clear what the president will discuss.

It comes as the Trump campaign and its allies have been seeking to overturn the results of the election in multiple states through lawsuits and intrusions into the state vote certification process, often targeting cities like Detroit, Philadelphia, Milwaukee and Atlanta with large and politically powerful Black populations. Mr. Trump himself reached out personally to at least one election official in Wayne County, Mich., home of Detroit, who tried to decertify the results there.

At the same time, he has made few public appearances since the election and his daily schedule often has no events on it, despite the worsening coronavirus pandemic.

Some members of Mr. Trump’s team have promoted the legally dubious theory that friendly legislatures could under certain scenarios effectively subvert the popular vote and send their own, pro-Trump delegations to the Electoral College.

The Trump campaign’s lead election lawyer, Rudolph W. Giuliani, announced Thursday morning that the campaign was withdrawing a federal suit it had filed seeking to stop the certification of results in Wayne County. The campaign attached the affidavits to the dismissal notice.

Mr. Biden won nearly 95 percent of the vote in Detroit and around 70 percent of the vote in Wayne County en route to winning Michigan by more than 150,000 votes.

With the withdrawal of the Wayne County suit, the Trump campaign and its Republican supporters have now lost or withdrawn from all of their major legal actions in Michigan, although the state’s Supreme Court is still considering an appeal of a lower court’s decision not to halt the certification of Wayne County’s results.

At a news briefing on Thursday, Gov. Gretchen Whitmer of Michigan, a Democrat, was asked about what her message to Mr. Trump would be on his election efforts.

“Stop spending energy to mislead about what happened in this election and spend it on a real Covid relief package,” she said. “This election was overwhelmingly decided. It was a safe, it was a secure, it was a fair election, and Joe Biden won the state of Michigan by over 150,000 votes.” She added, “The canvassers need to do their job. I expect that they will do their job and certify this result.”

Nov. 19, 2020, 10:00 a.m. ET

Nov. 19, 2020, 10:00 a.m. ET

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Sidney Powell, an election lawyer for President Trump, leaving a federal court last year with Michael T. Flynn.
Credit…Manuel Balce Ceneta/Associated Press

Sidney Powell, a lawyer on President Trump’s election legal team who represented the former national security adviser Michael T. Flynn, has been a major source and promoter of viral conspiracy theories about vote switching.

Since the election, Ms. Powell has advanced claims of voluminous voter fraud and a rigged election. She falsely claimed that a supercomputer called Hammer hacked votes, that Mr. Trump won the election by “millions of votes” and that voting software company Dominion Voting Systems altered the tallies.

Last week, she promised that coming evidence would overturn the election’s results and said she would “release the Kraken,” a reference to the 1981 movie “The Clash of the Titans,” reprising a catchphrase that began trending on Twitter.

On Monday, Ms. Powell posted some of her so-called evidence on Twitter. It consisted of three screenshots of an affidavit that she said was signed by a former military official from Venezuela about elections there. The screenshots were incomplete and did not include a name or signature, and Ms. Powell did not respond to requests to view the full document.

But according to her and excerpts from the affidavit, the elections software company Smartmatic helped the Venezuelan government rig its elections by switching votes and leaving no trail. The military official said in the excerpts that the U.S. election was “eerily reminiscent” of what happened in Venezuela’s 2013 presidential election, though no evidence was provided that votes had been switched in the United States.

Ms. Powell promoted the affidavit and its claims in interviews on conservative media that have amassed at least four million views on YouTube.

“This person saw, by his own experience, exactly what was happening there was happening here,” Ms. Powell explained to Fox News on Monday.

Smartmatic does not provide technology to the battleground states that sealed President-elect Joseph R. Biden Jr.’s victory. And electronic voting security experts said they were unimpressed with what Ms. Powell presented.

“The essence of the affidavit is that voting machines could have been hacked. This is not news,” said David Dill, a computer scientist at Stanford University and founder of the Verified Voting Foundation. “Every single vote that has been counted by computer in the U.S. in the last 50 years was counted by a computer that ‘could have been hacked.’ So far as I know, none of them actually were.”

Dan Wallach, a professor of computer science at Rice University and an expert on electronic voting system security, said: “If this class of attack was happening, the odds of it going undetected is quite low. So far, we have no evidence suggesting an abnormal number of spoiled ballots.”

Previous claims that Smartmatic’s voting machines were rigged in Venezuela have been disputed and are “unsubstantiated,” according to The Associated Press. It’s worth noting that Smartmatic accused the Venezuelan government of election fraud in 2017, pointing out that its machines were used when the opposition party won a majority in the country’s National Assembly in 2015.

The excerpts from Ms. Powell also included numerous inaccurate claims to imply a similarity between Venezuela’s elections and the U.S. election, chiefly drawing dubious parallels between Smartmatic and Dominion, which was used in several key states. Ms. Powell took the claims one step further, telling the right-wing media company Newsmax that Venezuela’s vote counting system was then “exported” to the United States.

The official she cited also said that Dominion’s system was “a descendant” of Smartmatic’s system, that they “did business together” and that Mr. Biden had overtaken Mr. Trump only when “vote counting was stopped.”

Smartmatic and Dominion have denied any exchange of technology and maintain that they are competitors. Dominion bought assets from a company three years after Smartmatic sold the company. And Mr. Biden overtook Mr. Trump in Pennsylvania and Georgia after days of consistent counting, while maintaining a lead in Arizona that narrowed as tallying continued.

The official also claimed to Ms. Powell that voting machines display and print out a paper ballot showing the results the voter intended, while the software itself “changes the information electronically.” But Dominion’s system does not work like that.

“The process that we see happening in Georgia and elsewhere that use similar ‘ballot marking device’ systems,” Mr. Wallach said, “is that the voter selects their choices with a computer of some sort, which then prints their ballot. The voter then typically carries that ballot to a ballot box, often with a scanner on top, and deposits the ballot.”

Any difference or attack on the tabulation system would be caught in postelection recounts, and “so far, none of them have caught anything other than human errors in the tabulation process, such as forgetting to load a memory card,” he said.

Mr. Dill said: “Courts demand strong evidence to overturn an election. From that perspective, this affidavit does not help make a case.”

Nov. 18, 2020, 5:26 p.m. ET

Nov. 18, 2020, 5:26 p.m. ET

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Last week, the Trump campaign published a series of posts on Facebook and Twitter identifying dead Americans whose names, the campaign alleged, were used to cast votes in this month’s election. The seven people were from Georgia and Pennsylvania, two battleground states that were crucial to Joseph R. Biden Jr.’s victory.

At least three of them, however, either did not actually vote in the election or were alive and well and cast legal votes, according to state and county election officials.

The name that spread the most online was Deborah Jean Christiansen of Roswell, Ga. On Facebook, 166 posts mentioning her name as proof of voter fraud collected over 280,000 likes, shares and comments from last Wednesday through Sunday, according to CrowdTangle, a Facebook-owned social media analytics tool. The vast majority of that activity came from a video post from the account for “Tucker Carlson Tonight,” the Fox News show. The post, “Yes, Dead People Did Vote in the Election,” generated 2.5 million views on Facebook.

But Ms. Christiansen did not vote, according to election officials.

“We don’t have a record of a new voter registration, and we don’t have a record of a ballot being sent to this person,” Jessica Corbitt, a spokeswoman for Fulton County in Georgia, said in an interview. “We have her in the system as deceased.”

Some news outlets, like CNN and Agence France-Presse, reported that there was no fraud in Ms. Christiansen’s case. But each of the posts generated far fewer shares and interactions than the posts containing the false information, according to CrowdTangle data.

The Trump campaign also argued that James E. Blalock Jr. of Covington, Ga., and Linda Kesler of Nicholson, Ga., had voted fraudulently. But county election officials told The New York Times that the two people had been correctly marked as deceased and did not vote. Mrs. James E. Blalock Jr., the widow of Mr. Blalock, and a Lynda Kesler with a different address, birthday and Social Security number, voted legally, the officials said.

The Trump campaign’s original posts about Mr. Blalock and Mrs. Kesler collected 26,600 likes and shares on Facebook, according to CrowdTangle data, while a report from a local news outlet correcting the claim collected just 10,100.

The post about Mr. Blalock was eventually deleted on Twitter but remains up on Facebook. On Friday, Mr. Carlson apologized on air for his erroneous reporting in the case of Mr. Blalock.

“On Friday, we began to learn some of the specific dead voters reported to us as deceased are in fact alive,” Mr. Carlson said in a statement on Tuesday evening. “We initially corrected this on Friday. We regret not catching it earlier. But the truth remains: Dead people voted in the election.”

The other four people the Trump campaign held up are from Trenton, Ga., and Drexel Hill, South Park and Allentown, Pa. Local election officials said they were still investigating those allegations.

The Trump campaign did not immediately respond to a request for comment.

Election workers counting absentee ballots in Detroit earlier this month. 
Credit…Brittany Greeson for The New York Times

For a few hours on Tuesday, it looked as though two Republican officials in Wayne County, Mich., might reject the will of hundreds of thousands of voters.

The election canvassing board in Wayne County — a largely Democratic area that includes Detroit — met to certify the results of the Nov. 3 election and deadlocked along party lines, with the board’s two Democrats voting to certify and its two Republicans voting not to.

The Republican members, William Hartmann and Monica Palmer, said they were concerned about small discrepancies between the number of votes cast in some precincts and the number of people precinct officials recorded as having voted.

But these sorts of inconsistencies are not out of the ordinary. They can happen if, for instance, a voter checks in but then gets frustrated by a long line and leaves.

They were nowhere near significant enough in Wayne County, or anywhere else in Michigan, to change President-elect Joseph R. Biden Jr.’s victory. Mayor Mike Duggan of Detroit said they involved just 357 votes out of about 250,000 cast in the city.

Election certification is supposed to be routine: Canvassers at the county or municipal level (depending on the state) review precinct results, make sure every ballot is accounted for and every vote was counted, double-check the totals and send the certified numbers to state officials. It’s the process by which the results reported on election night are confirmed.

This is basically an accounting task. If the canvassers find possible errors, it is their job to look into and resolve them, but refusing to certify results based on minor discrepancies is not normal. Michigan’s canvassing boards always have four members split between the two parties, and it is extremely rare for members to decline to certify an election that their party lost.

“It is common for some precincts in Michigan and across the country to be out of balance by a small number of votes, especially when turnout is high,” Jocelyn Benson, the Michigan secretary of state, said in a statement Tuesday evening. “Importantly, this is not an indication that any votes were improperly cast or counted.”

It is also highly abnormal to suggest, as Ms. Palmer did, that canvassers certify the results in one place but not another when there is no meaningful difference between the two in terms of the number or severity of discrepancies.

Before the deadlock was resolved, Ms. Palmer had proposed certifying the results in “the communities other than the city of Detroit.” As Democrats and election law experts noted, nearly 80 percent of Detroit residents are Black. By contrast, in Livonia — a city west of Detroit that had the second-highest number of discrepancies in the county, but whose results Ms. Palmer was willing to certify — less than 5 percent of the population is Black.

After intense backlash, both from election watchdogs and from voters whom Representatives Debbie Dingell and Rashida Tlaib organized to call in to the canvassing board’s meeting, Mr. Hartmann and Ms. Palmer voted to certify the results after all. While they demanded that Ms. Benson conduct an audit of the Wayne County results, that will not delay the certification process.

By Wednesday morning, every county in Michigan had certified its results. The Board of State Canvassers will meet on Nov. 23 to certify the statewide totals.

Nov. 18, 2020, 4:05 p.m. ET

Nov. 18, 2020, 4:05 p.m. ET

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A frame grab from a video when President Trump said falsely, “If you count all the legal votes, I easily win.” It was posted to Mr. Trump’s YouTube page with two million subscribers.
Credit…YouTube

YouTube videos endorsing the false idea that there was widespread election fraud were viewed more than 138 million times on the week of Nov. 3, according to a report from an independent research project that has been studying misinformation trends on the video site.

The report by the project, called Transparency.tube, looked at videos on YouTube that supported claims of voter fraud during the November elections, as well as videos that disputed such claims. Over all, the researchers identified 4,865 videos, viewed a combined 409 million times, that mentioned voter fraud.

The YouTube videos supporting claims of voter fraud accounted for 34 percent of all views in the data set studied, while those disputing the voter fraud claims or remaining neutral accounted for 66 percent of views among the videos the research project identified.

YouTube does not release data about the total number of videos uploaded to the site weekly. The company has said that 500 hours of video are uploaded to YouTube every minute.

Many of the largest YouTube channels can rack up millions of views each day. For example, CNN, which has over 11 million subscribers to its YouTube channel, uploaded 51 videos during the week of Nov. 3. Those videos were viewed 69 million times, according to an analysis by The New York Times.

Some of the most-watched videos disputing the results of the election include two videos by the right-wing news outlet BlazeTV, which were viewed 1.3 million times. Videos by the right-wing news outlets Newsmax and OANN that spread claims of widespread voter fraud were also viewed hundreds of thousands of times.

False claims, as varied as reports of malfunctioning voting machines and intentionally miscounted mail-in votes, have been widely circulated on all social media platforms, including YouTube. Election officials and journalists investigating voter fraud have found no evidence for claims of widespread voter fraud.

YouTube has said that videos disputing voter fraud allegations were more widely viewed on its platform than those supporting it, but has declined to give numbers.

“The most-viewed videos related to ‘voter fraud’ are all from authoritative news channels and the majority of election-related searches and recommendations are surfacing results from authoritative sources,” said Farshad Shadloo, a YouTube spokesman. He added that panels linking to a “Rumor Control” page debunking potential areas of election-related misinformation were shown billions of times.

The researchers behind the Transparency.tube report said YouTube’s statements told only part of the story. The report, the researchers said, showed that not only were significant numbers of people watching videos filled with misinformation about voter fraud, they were reaching those videos even though YouTube was directing them elsewhere.

“Videos supporting accusations of widespread voter fraud have been popular despite YouTube’s video recommendations,” said Mark Ledwich, a co-founder of Transparency.tube. He said that people were arriving at videos filled with misinformation through “direct links, channel subscriptions or search.”

Nov. 18, 2020, 10:31 a.m. ET

Nov. 18, 2020, 10:31 a.m. ET

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Rudolph W. Giuliani announced that the Trump campaign would file suits challenging the vote count in Pennsylvania the day after the election.
Credit…Gabriela Bhaskar for The New York Times

Rudolph W. Giuliani, President Trump’s personal lawyer, has spread a litany of falsehoods and conspiracy theories in media appearances and social media over the past week.

Mr. Giuliani, who has a long history of fudging the truth and who has led the Trump campaign’s largely unsuccessful legal fight over the election, has focused particularly on debunked claims of barred poll workers and unsubstantiated conspiracy theories about a voting software company affecting the election’s outcome.

In interviews on Fox News, Mr. Giuliani has repeatedly claimed that Democratic officials blocked Republican poll watchers from observing ballot counting in “10 different crooked Democratic cities,” including Philadelphia, Pittsburgh, Detroit, Milwaukee, Reno, Phoenix and Atlanta. And in the counties where Philadelphia and Pittsburgh are, he has said, the lack of access affected over 680,000 votes.

There’s no evidence to support any of these allegations. Mr. Trump’s own legal filings acknowledged the presence of Republican observers in Nevada, Pennsylvania, Georgia and Arizona and there were at least 134 Republican poll challengers present inside TCF Center in Detroit, a convention center where votes were counted.

Mr. Giuliani has brought up Philadelphia and Pittsburgh several times. That’s because a Trump campaign lawsuit had claimed that some 682,000 ballots in those cities’ two counties were processed “when no observation was allowed” and sought to have those votes thrown out.

But as The Washington Post and The Associated Press have reported, and as Mr. Giuliani said in court on Tuesday, the Trump campaign recently revised its lawsuit and withdrew this particular claim.

“My judgment is that when Hillary Clinton said to Biden about four weeks ago don’t concede no matter what, she meant even if you’re behind by 800,000 votes in Pennsylvania, Joe, don’t worry, we’ll fix it for you,” Mr. Giuliani misleadingly said on Nov. 9. (Mrs. Clinton did not say Mr. Biden should “never” concede, but rather he shouldn’t on election night because counting mail-in ballots could “drag out” for days.)

Mr. Giuliani has also accused Dominion Voting Systems, a voting software company, of having foreign, and seemingly nefarious ties.

The company, he has said in several Fox News appearances, is associated with those who were “very close” to two Venezuelan presidents, Hugo Chávez and Nicolás Maduro, as well as the billionaire financier George Soros, because it is “owned by another company called Smartmatic.”

That is wrong. Smartmatic has said it has never owned shares, had financial stakes or provided software or technology to Dominion. Dominion’s chief executive said in an April letter to Congress that he owned a 12 percent stake of the company, while a private equity firm, Staple Street Capital Group in New York, owned about 76 percent. (No other investor held more than 5 percent of Dominion.)

Smartmatic previously owned a voting machine company, Sequoia Voting Systems, before selling it in 2007, as the Washington Post Fact Checker reported. Dominion purchased some assets from Sequoia Voting Systems in 2010.

Smartmatic itself was founded in Florida, incorporated in the United States and based in London. It overhauled Venezuela’s election machinery in 2004 and took out a loan from the country. In the 2017 election there, the company said the Venezuelan government falsified turnout figures. That led the government to reject its claims and threaten legal action, undermining Mr. Giuliani’s claims that Smartmatic was “close” with Mr. Maduro.

Smartmatic’s connection to Mr. Soros is similarly tenuous. Its chairman, Mark Malloch-Brown, sits on the board of Mr. Soros’s Open Society Foundations and, Smartmatic notes, a dozen other organizations.

Nov. 18, 2020, 10:07 a.m. ET

Nov. 18, 2020, 10:07 a.m. ET

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Representative Louie Gohmert in Philadelphia while votes were being counted there on Nov. 6. He has repeated false claims that the Army seized an election-related computer server.
Credit…Spencer Platt/Getty Images

Allies of President Trump are spreading another baseless rumor about computer-based vote manipulation, days after they gained attention for falsely claiming that a widely used piece of election administration software had been used to delete votes for the president.

The newest conspiracy theory involves Scytl, a software company in Barcelona, Spain, that makes software for local election officials.

The false theory alleges that the U.S. Army recently raided Scytl’s office in Frankfurt and seized a computer server containing authentic vote totals for the 2020 election. This “undoctored” data, the theory claims, shows that Mr. Trump was not defeated but instead won in a landslide with 410 electoral votes.

Both Scytl and the Army have refuted the claim. An Army spokesperson told The Associated Press that there had been no raid on Scytl’s offices and no servers seized. In a fact-check posted to its website, Scytl said it did not “tabulate, tally or count votes” in U.S. elections or have an office in Frankfurt.

Jonathan Brill, the president and general manager of Scytl’s U.S. division, said in an interview on Tuesday that the rumor that the company’s software had been used to tamper with vote tallies was “totally false, every single bit of it.”

The false claim appears to have originated with a Twitter post on Nov. 8 by a user, @zeynep_mol, who claimed to have heard about the raid. (“I haven’t been able to confirm the accuracy yet,” the account tweeted.) The story was then picked up by a little-known Indian news website, GreatGameIndia, which gained notoriety this year for spreading false claims about the origin of the coronavirus pandemic.

It was then repeated by Representative Louie Gohmert, Republican of Texas, during interviews with Newsmax, the conservative TV network, and Charlie Kirk, the right-wing activist, and has since been shared by other prominent conservatives hoping to cast doubt on the election outcome.

Scytl, which was started in Spain in 2001, does make software for local election officials, including some in the United States. In the 2020 election, it says, it provided four types of products to local authorities. One is a system that allows election officials to display results from their elections in a user-friendly format. Another product, “electronic ballot delivery,” helps local election officials deliver ballots to absentee voters.

But the company says none of its products are used to count votes, or allow voters to vote online.

Some people who have shared the Scytl theory have alleged that the company has ties to George Soros or Bill Gates, two billionaire philanthropists who are often featured in right-wing conspiracy theories.

Mr. Brill, the president of Scytl’s U.S. division, said there was no truth to those rumors, either.

“We have no investment from George Soros nor Bill Gates,” he said.

Nov. 17, 2020, 4:09 p.m. ET

Nov. 17, 2020, 4:09 p.m. ET

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Supporters of Joe Biden celebrated in downtown Austin on Nov. 7 hours after it was announced that he had won the presidency.
Credit…Tamir Kalifa for The New York Times

In recent days, conservative websites have reported that a high-level staff member in Texas for the election campaign of President-elect Joseph R. Biden Jr. was arrested on charges of voter fraud for his role in supposedly helping to orchestrate a ballot harvesting scheme.

The news seemed to acquire additional credibility when supporters of President Trump began circulating a photo purporting to show the staff member, Dallas Jones, in handcuffs being escorted by police officers.

But Mr. Jones, a political consultant based in Houston who serves as the Texas state political director for the Biden campaign, called the rumors “lies.”

“I was not arrested,” he said Tuesday in a phone interview. “These are categorically made-up allegations. They are baseless.”

There have also been no voter fraud criminal cases filed against Mr. Jones, according to court records in Harris County, Texas, where Houston is located.

An official with the Biden campaign also denied additional reports that Mr. Jones had been fired from his campaign job, calling the claims surrounding Mr. Jones “laughably false.”

As for the photo, it was from June 2019 and shows the actor Cuba Gooding Jr. after he surrendered to police in New York and was charged with sexual abuse and forcible touching after a woman accused him of groping her at a rooftop bar.

The rumors surrounding Mr. Jones stem from a court case initiated by Republicans in September that sought to block the extension of time for mail-in and early voting in Harris County. The Texas Supreme Court rejected the Republican request in October.

The court filings included accusations of an illegal “ballot harvesting operation” in Harris County, where volunteers and other workers were gathering absentee and mail-in ballots and delivering them to ballot collection sites. Citing inquiries conducted by private investigators, the filings claimed that the “harvesters” were taking the ballots of homeless people and nursing home residents, forging signatures and picking candidates of their own choosing.

According to the court filings, the investigators accused Mr. Jones, among others, of helping to coordinate the operation.

In a response filed in the case, Chris Hollins, the Harris County clerk, called the claims “wild” and “baseless.”

“These accusations are backed by no evidence whatsoever and are of dubious provenance, coming from private investigators who do not disclose either their clients or sources,” he wrote.

Nov. 17, 2020, 3:08 p.m. ET

Nov. 17, 2020, 3:08 p.m. ET

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People waited in line on Friday to address ballot issues and questions at the Clark County Election Department in Las Vegas this month.
Credit…Bridget Bennett for The New York Times

President Trump claimed a “big victory” on Monday night after officials in Clark County, Nev., voted to rerun one local race that had a margin of just 10 votes.

In the president’s words, officials found “large-scale voter discrepancy” and “did not have confidence in their own election security.”

In fact, the news in Clark County was bad for the president; a review of the votes found a relatively tiny number of problems; and the campaign of the Republican in the close race repudiated the president’s claims about the election’s security.

The most significant news for President Trump in Nevada on Monday was that officials in Clark County, by far the state’s most populous county, certified their presidential election results, confirming that the president lost by more than 90,000 votes in the county.

In a so-called canvass of the county’s results, in which election officials check the vote counts against other voting records, officials found 936 discrepancies, or less than 0.1 percent of the 974,185 votes in Clark County. That rate will not affect the president’s deficit of more than 33,000 votes in Nevada. President Trump’s statewide deficit is smaller than his gap in Clark County because he won Nevada’s rural counties.

Clark County commissioners voted on Monday to certify the results of all of the county’s elections except one: a race for the county commission seat that represents District C, an area with 332,000 residents that includes parts of Las Vegas. The margin in that race was just 10 votes. In that district, election officials found 139 discrepancies, in line with the .09 percent rate of discrepancies to votes countywide. The county commission called for a special election.

“There is no election that goes without discrepancies,” Joe Gloria, Clark County’s registrar of voters, told county commissioners on Monday. With just 10 votes separating the two candidates, “it’s very difficult to get through that without having the discrepancies being larger than the margin of victory.”

Discrepancies in Clark County included cases in which signatures on mail-in ballots could not be verified or in which voter check-in books at polling places did not align with the number of votes cast there, often the result of human error, Mr. Gloria said.

“When we find discrepancies, we report them, and when the number of discrepancies is larger than the margin of victory, we hold a runoff election,” the Clark County Election Department said in a statement. “As the voters can see, this issue did not affect any other outcomes, and to claim otherwise is simply untrue.”

There was no sign of “large-scale” voter discrepancies, said Lisa Mayo-DeRiso, campaign manager for Stavros Anthony, the Republican, who will now compete in the special election for the Clark County commission seat against Ross Miller, the Democrat, who led the first election by 10 votes.

“It’s not an accurate statement,” she said of the president’s claim that Clark County officials did not have confidence in the election’s security. “We have not had any problems or difficulties with the Clark County Election Department.”

Nov. 17, 2020, 1:51 p.m. ET

Nov. 17, 2020, 1:51 p.m. ET

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Sidney Powell, left, a lawyer for the former national security adviser Michael T. Flynn, used the phrase last week. 
Credit…Manuel Balce Ceneta/Associated Press

A catchphrase winking at an unfounded voter fraud conspiracy in the presidential election trended on Twitter on Tuesday morning as President Trump’s allies continued to contest the outcome of the election, which has been called for Joseph R. Biden Jr.

The phrase, “Release the Kraken,” appeared on Twitter’s trending topics list on Tuesday, collecting nearly 100,000 tweets, pushed mostly by conservatives and far-right internet personalities. It is a catchphrase from the 1981 movie “The Clash of the Titans,” but this time it was used to signal an election fraud conspiracy on social media.

The conspiracy stems from a Fox Business Network appearance last Friday by Sidney Powell, a lawyer for the former national security adviser Michael T. Flynn. During an interview with the host Lou Dobbs, Ms. Powell claimed that the president’s team had voluminous evidence that it planned to release to overturn election results in key states.

“We are talking about hundreds of thousands of votes,” Ms. Powell said in the interview. “President Trump won this election in a landslide.”

Ms. Powell has not yet provided any evidence and did not respond to a request for comment. The New York Times has called election officials in every U.S. state and found no evidence of voter fraud.

At a later point in the interview, Ms. Powell asserted that the voter fraud had been “organized and conducted with the help of Silicon Valley people, the big tech companies, the social media companies and even the media companies.”

“I’m going to release the Kraken,” Ms. Powell said.

The video gathered steam over the weekend and began to trend on Twitter by Tuesday morning. One video of the television appearance, which was posted to YouTube on Saturday, gained 1.3 million views in four days. The views were driven by over 160,000 likes and shares on Facebook, 81 percent of which happened in private Facebook groups and on users’ private feeds according to data from CrowdTangle, a Facebook-owned social media analytics tool.

Twitter and Facebook did not immediately comment. YouTube said the video did not violate its guidelines and that it was showing an information panel about election results under the video.

The false narrative has also been promoted by far right personalities with track records of spreading misinformation, such as David J. Harris Jr., who has shared false conspiracies about the coronavirus, and Austen Fletcher, a right-wing internet personality who spread false rumors of voter fraud in Michigan that in reality could be attributed to run-of-the-mill clerical errors.

Mr. Harris did not respond. When asked for comment, Mr. Fletcher said: “Sidney Powell is a brilliant woman… When ‘the kraken’ is released America will remember who sought the truth and who asked no questions.”

Recounting ballots by hand in Marietta, Ga., on Sunday.
Credit…John Amis/Atlanta Journal-Constitution, via Associated Press

As Georgia election officials work to complete a recount of nearly five million ballots by Wednesday, President Trump is trying to cast doubt on the legitimacy of the process — a hand recount that his own campaign demanded.

In a series of tweets in recent days, Mr. Trump has alleged that vote counters have not been permitted to match signatures included with absentee ballots and has falsely claimed that this renders the recount meaningless. He has also claimed incorrectly that the inability to match the signatures is the result of a consent decree signed by the state.

Here are the facts:

In Georgia, where the initial count showed that Mr. Trump lost to President-elect Joseph R. Biden Jr. by several thousand votes, the recount process does not include the reverification of signatures included with absentee ballots.

When absentee ballots are received by Georgia’s election officials, the signature on the envelope is matched to other signatures that are part of the voter’s record. Once that is verified, the envelope containing the signature is separated from the ballot to protect the secrecy of the voter’s choice. Voters whose signatures do not match those on record are notified and asked for clarification.

The envelopes and ballots are retained for two years. But because they have been separated to protect voters’ privacy, there is no longer a way to match ballots to envelopes. As such, rechecking signatures in a recount would be meaningless.

A similar procedure is followed in other states, and Georgia’s process of separating ballots from signature envelopes has nothing to do with any consent decree.

Georgia’s secretary of state, Brad Raffensperger, a Republican, has said that the state trained election officials on signature matching. More than 1.3 million people voted by absentee ballot in Georgia in the general election.

Mr. Raffensperger also has accused fellow Republicans of trying to undermine the legitimacy of the state’s election in an effort to swing the results to Mr. Trump.

In an interview with The Washington Post on Monday, Mr. Raffensperger said that Senator Lindsey Graham of South Carolina had asked him about possible ways that ballots could be disqualified, including whether the secretary of state could reject all absentee ballots in counties that had a high number of signature mismatches.

Mr. Graham, speaking to reporters on Capitol Hill on Monday, denied that he had suggested to Mr. Raffensperger that he find a way to throw out legally cast ballots. He said that his conversation with the secretary of state covered questions about Georgia’s system of verifying signatures with absentee ballots.

Election officials in Georgia also announced Monday evening that they had discovered 2,600 ballots in Floyd County that had not been previously reported to the state, an error they attributed to “gross incompetence” on the part of the county’s elections director. The newly discovered ballots will reduce Mr. Biden’s margin by a few hundred votes, they said — not enough to put his victory in jeopardy.

Mark Zuckerberg of Facebook and Jack Dorsey of Twitter testified on Tuesday about their platforms, misinformation and the 2020 election.

Mr. Zuckerberg said that while many people on the social network disliked the presence of misinformation, he also strongly believed that Facebook should not be the arbiter of what content is or isn’t true. Mr. Dorsey said he wanted to give users more control over the algorithms that moderate content on Twitter.

Nov. 17, 2020, 9:58 a.m. ET

Nov. 17, 2020, 9:58 a.m. ET

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President Trump’s supporters protested the election’s results in Phoenix two days after the election.
Credit…Adriana Zehbrauskas for The New York Times

A prominent electoral map on the website of Everylegalvote.com, a self-described fraud-buster, conveyed an alternate reality on Monday.

Despite the latest election results showing Joseph R. Biden Jr. winning a decisive 306 votes in the Electoral College compared with 232 for President Trump, the site’s map showed Mr. Trump as the winner of the election.

Mr. Trump had received 232 votes compared with 214 for Mr. Biden, according to the site’s map, which was flecked with orange to connote states where it claimed that voter fraud had been detected. Click on a tab saying “without voter fraud,” and Mr. Trump’s vote suddenly leapt to 318 against Mr. Biden’s 220.

As President Trump refuses to concede the election, a lot of internet traffic is being directed to this slickly produced website channeling the president’s mix of falsehoods, conspiracy theories and baseless accusations of voter fraud.

The website has promoted the false narrative that mail-in ballots were used to steal the election from Mr. Trump. It has posited that thousands of dead people voted in Michigan when they did not. It has also posted content from a source with links to QAnon, the elaborate conspiracy movement that falsely claims the existence of a Satan-worshipping pedophile cabal run by senior Democrats, who are plotting against Mr. Trump. QAnon believers had predicted that Mr. Trump would easily win the election.

“They are attempting to install Joe Biden as president without due process of law and order,” the site says, citing the media, “certain elected officials” and people in positions of power, both in the United States and in other countries.

A review of Everylegalvote.com shows an attempt to delegitimize the election under a veneer of empiricism by drawing on murky and debunked theories. The site did not respond to a request for comment.

Everylegalvote.com lists its sponsors and financial backers as a coalition of right-wing groups including Allied Security Operations Group, the Economic War Room, and Liberty Center for God and Country.

An office manager at Allied Security Operations Group, who declined to give her name, said by phone that the group was a private Dallas-based cybersecurity firm. She said the issue of voter fraud was an area of specialty of the company’s chief financial officer, Russell Ramsland, a businessman who ran for Congress in Texas as a Republican in 2016 and was defeated in the primary. According to Mr. Ramsland’s LinkedIn profile, he has an M.B.A. from Harvard.

The Economic War Room listed on Everylegalvote.com’s website appears to refer to the “Economic War Room With Kevin Freeman,” a weekly television financial news show. Mr. Freeman was not immediately available for comment.

Mr. Freeman is described on Economic War Room’s website as “one of the world’s leading experts on the issues of economic warfare and financial terrorism.” He is also listed as a senior fellow at the Center for Security Policy, a Washington-based think tank that has been designated as a hate group with anti-Muslim ideology by the Southern Poverty Law Center.

Last year, Mar-a-Lago, President Trump’s private club in Palm Beach, Fla., came under criticism after agreeing to host an event by the group.

In a broadcast in the days after the election, which was posted on Economic War Room’s website, Mr. Freeman called the election “one of the most contested elections in history.” “The left is out for blood and total victory,” he said.

He predicted a violent insurrection by the “far left” and said it wanted to tear down God and family and install a nanny state. As evidence, he cited attempts in states like California to curtail family gatherings during Thanksgiving and make people wear masks in between bites of turkey.

Liberty Center for God and Country, the site’s other backer, said it aimed to “promote and protect our God-given, unalienable Constitutional rights and liberties” and to “support legal efforts that protect these liberties.” It recently posted on its website a request by President Trump for every registered voter to send him a handwritten letter, demanding “a full election audit of all 50 states.” The group was not immediately available for comment.

Everylegalvote.com has not shied away from promoting misinformation from dubious sources.

After President Trump made baseless claims that Dominion Voting Systems, which makes software that local governments use to help run their elections, had software glitches that changed voting tallies in key states, the site posted a Twitter thread on the software maker by Ron Watkins, who has been part of QAnon’s inner circle.

In his thread, Mr. Watkins posited that local information technology “guys” could have hacked into the system to alter election results.

A group of federal, state and local election officials have said “there is no evidence” any voting systems were compromised.

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