JUDY WOODRUFF: Next tonight, political turmoil in Russia and more challenges to Vladimir Putin.
Margaret Warner has the story.
MARGARET WARNER: Freezing temperatures did not deter the determined crowds protesting in Moscow on Saturday. Police put the gathering at 30,000, organizers at four times that many.
It was the second round of mass rallies across Russia this month, fueled by charges that the ruling United Russia party cheated to win the Dec. 4 parliamentary elections.
NATALYA SOLOVYOVA, protester (through translator): I'm enraged by the falsifications. They actually stole the votes of people who didn't vote for them. I got angry, and that is why I'm here.
MARGARET WARNER: Blogger Alexei Navalny was detained after the first round of protests, but he was back by Saturday vowing, "If these crooks and thieves keep cheating us, we will take what is ours."
ALEXEI NAVALNY, blogger (through translator): It is an obvious machination, obvious fraud done in such a brazen way that it is impossible to tolerate this any longer. I think at least 150,000 people gathered here despite the cold temperature. But even if it were minus-20, the number would be the same.
MARGARET WARNER: The target of the crowd's ire? United Russia's man Vladimir Putin, who has held power for 12 years, first as president and now as prime minister.
Former finance minister Alexei Kudrin, who still talks to Putin, called for the new parliament to be dissolved and face new elections.
ALEXEI KUDRIN, former Russian prime minister (through translator): I support snap elections. I'm ready to take part and facilitate this dialogue in order to work out the solutions.
MARGARET WARNER: But at United Russia's headquarters today, Putin rejected the idea of a new vote and sounded dismissive of the protesters.
VLADIMIR PUTIN, Russian prime minister (through translator): The problem is they lack a consolidated program, as well as clear and comprehensible ways of achieving their goals, which aren't clear either. They also lack people who are capable of doing something concrete.
MARGARET WARNER: He did call for reforms to govern future elections, however. And he pledged a clean process in the presidential balloting in March, when he hopes to regain his old office.
VLADIMIR PUTIN (through translator): As for me, as a candidate, I don't need any vote-rigging. I want the election to be maximally transparent. I want everybody to understand it.
MARGARET WARNER: Current President Dmitry Medvedev also proposed political reforms in his final State of the Union speech last Thursday. He and Putin catalyzed Russian discontent by announcing in September that they planned to switch jobs again.
Then came the allegations of election fraud and days of angry protests. Saturday's demonstrations were the largest yet since those that brought down the Soviet Union, which was dissolved 20 years ago this week.
Former Soviet leader Mikhail Gorbachev, who resigned on Christmas Day, 1991, called for Putin to do the same during an interview on Saturday.
MIKHAIL GORBACHEV, former Soviet president (through translator): I would advise Vladimir Putin to step down now. It has been enough, three terms, two terms as president and another term as prime minister. Three terms, that's enough.
MARGARET WARNER: For now, though, Putin shows no sign of taking that advice. He's pointing toward the March 4 presidential election day, when he will need at least 50 percent of the Russian vote to avoid a runoff.