Sen. Lindsey O. Graham agreed to hear Rudolph W. Giuliani out.

In a Jan. 2 meeting arranged by Mark Meadows, the White House chief of staff, and held in his West Wing office, the South Carolina Republican met with Giuliani and his legal team to learn about findings they said could hand Donald Trump a second term.

Giuliani, Trump’s personal attorney at the time, put forward a computer whiz who presented a mathematical formula suggesting Biden’s support in certain states was unrealistic. Graham, a lawyer and chairman of the Senate Judiciary Committee, found the reasoning too abstract. He wanted hard evidence. “Give me some names,” Graham said at the Saturday meeting. “You need to put it in writing. You need to show me the evidence.”

Giuliani promised details by Monday — proof that scores of ballots had been cast in the names of dead people and people under 18, among other irregularities.

This scene is recounted in a new book by Washington Post associate editor Bob Woodward and national political reporter Robert Costa. The book, “Peril,” describes parallel efforts by the South Carolina Republican and his conservative colleague from Utah, Sen. Mike Lee, to personally investigate the president’s claims of voter fraud as the lawmakers prepared to certify Joe Biden’s victory on Jan. 6.

Graham and Lee, both of whom ultimately voted to certify the results, took the claims of election fraud seriously enough to get briefed on the details, involve their senior staff and call state officials throughout the country. But privately, Graham gave the arguments a withering assessment, according to the book, saying they were suitable for “third grade.”

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The episode illustrates how strenuously the president’s legal team sought to nullify the results of the election; how flimsy even their more serious claims were; and what little stock the president’s own allies placed in his objections, even as they stood steadfastly with their standard-bearer.

Giuliani’s promised proof arrived on Jan. 4 in the form of several memos he sent to Graham, one titled “Voting Irregularities, Impossibilities, and Illegalities in the 2020 General Election.”

Graham had already proved himself willing to act on assertions of election irregularities, calling Georgia’s secretary of state, Brad Raffensperger, in November to see if he could toss mail ballots in counties found to have high rates of nonmatching signatures, Raffensperger later said. Graham maintained he was only inquiring about the state’s signature-matching requirements.

The memos from Giuliani included sweeping claims lacking references or evidence. One said Pennsylvania had processed 682,777 mail-in ballots without proper observation — an assertion underlying a suit rejected by a federal judge two months earlier. “If you deduct just this number, President Trump wins the state by hundreds of thousands,” the memo argued. Another claimed hundreds of dead people had voted in Georgia, based on an extensive analysis of “mail-in and absentee ballot voter names and obituaries.”

Graham dispatched his driver to deliver the documents to his top lawyer on the Judiciary Committee. The lawyer, Lee Holmes, was unimpressed, according to the account in “Peril.”

It was impossible for him to tell precisely what kind of records had been used to conduct the analysis, which proved nothing conclusively, in his view. Of the more sensational claims, such as ballots from the deceased, Holmes thought it was much more likely, based on Giuliani’s own evidence, that some people had voted and then died, according to the book. He was equally unconvinced by theories about people voting twice, improper absentee ballot applications and fraudulent ballots cast from vacant or nonexistent addresses. “Holmes could find no public records that would even allow someone to reach these conclusions,” according to the account in “Peril.”

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Where he could find relevant records, they contradicted Giuliani’s conclusions. Claims of nearly 12,000 so-called “overvotes” in Arizona — when someone picks more than the maximum number of selections allowed — masked the fact that only 180 applied in the presidential contest, not nearly enough to close Biden’s margin of victory in the state.

The argument that “Georgia’s 2020 election is in doubt and should be nullified,” was laid out in an “Executive summary” of “Independent analysis conducted by expert CPAs and Ivy League statisticians,” according to the documents provided by Giuliani. The experts were unnamed, but their alleged probe — correlating data for 7.6 million registered voters with postal records — struck Holmes as unfeasible or, at best, inconclusive, because it would involve matching millions of registered voters with U.S. Postal Service data. Similar analysis failed to convince a District Court judge in Nevada, who wrote in a December judgment dismissing a suit from the Trump campaign that there was insufficient information about how the data was obtained and used and “what the rate of false positives would be.”

Graham’s attorney shared those concerns.

“Holmes found the sloppiness, the overbearing tone of certainty, and the inconsistencies disqualifying,” the authors write. The memos, he determined, “added up to nothing.”

But it wasn’t until after a pro-Trump mob ransacked the Capitol that Graham, speaking on the Senate floor, said, “Count me out. Enough is enough. I’ve tried to be helpful.”

He has since tacked back, visiting Trump at Mar-a-Lago, speaking to him regularly and saying the GOP “can’t grow” without the former president. Still, he has continued to deliver criticism directly to Trump, according to the book. In a phone call this summer, he bemoaned Trump’s volatility and focus on voter fraud, telling the former president, “You [obscenity] your presidency up.” Trump abruptly hung up on him. A spokesman for Graham declined to comment.

The same day Graham met with Giuliani in the West Wing, Lee received a two-page memo from the White House marked “PRIVILEGED AND CONFIDENTIAL.” It included a stunning claim — that Vice President Mike Pence could hand the election to Trump because seven states had submitted dueling slates of electors to Congress, split between the incumbent and Joe Biden. Pence could simply set those states aside on Jan. 6 and count only electors from the remaining states, it claimed. “Pence then gavels President Trump as re-elected.”

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That outcome was envisioned by John Eastman, the conservative legal scholar and author of the memo, titled “January 6 scenario,” which was obtained for “Peril” and reviewed by The Washington Post.

Lee knew dueling electors were merely Trump loyalists putting themselves forward in certain states, in a move the authors describe as “a social media campaign — an amateur push with no legal standing.” Electors are generally bound by the popular vote in each state; because they sum to 538, an absolute majority of 270 is needed to clinch the presidency.

The authors suggest the senator, a former law clerk for Supreme Court Justice Samuel Alito, was surprised this theory had been circulated by Eastman, a professor at the Chapman University School of Law and former law clerk to Justice Clarence Thomas. Document in hand, and bewildered that theories about dueling electors were still coming from Trump’s legal team, Lee made “phone call after phone call” to officials in some of the relevant states, such as Georgia, Pennsylvania and Arizona, he told constituents in a Jan. 27 online town hall, appearing to refer to the Eastman memo without naming its author. A spokesman for Lee did not respond to a request for comment.

No one seemed poised to certify a new slate of electors. “At that point, I believed that we had reached the end of the process, as indeed we had,” Lee said during the town hall.

The senator also explained his interpretation of the limited role the Constitution gave to Congress and the vice president in counting electoral votes — an interpretation in conflict with the one outlined by Eastman, who argued Pence could be the “ultimate arbiter” and either name Trump the president-elect or send the matter to the House.

Eastman even anticipated what “Peril” describes as “certain outrage and worry of a coup.” In the memo, however, he dismissed these concerns as, “Howls, of course, from the Democrats …”

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Four days after his memo reached Lee, the law professor addressed Trump supporters outside the White House on Jan. 6. A week after that, Chapman University announced his immediate retirement from the Orange, Calif., school.

In an email to The Post, Eastman said his memo merely “explored all options that had been proposed.” Ultimately, his advice to Pence was not to act on the basis of the dueling electors, he added, because “at that point, no legislature had certified an alternative slate of electors.” Rather, he counseled the vice president to delay certification, Eastman said, by pointing to objections leveled by state legislators — a move Pence explored, according to “Peril,” but did not make.

Eastman’s remarks on Jan. 6, he said, were unplanned and designed to “fill a gap in the roster of speakers when the president’s arrival was delayed.”

Lee, for his part, said in a Fox News interview a month after the Jan. 6 riot that Trump deserved a “mulligan” for the speech he delivered encouraging his supporters to march on the Capitol, using a golf term for giving someone another try.

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The Washington Post’s Claire McMullen contributed to this report.