By Guy Faulconbridge and Alexei Anishchuk
MOSCOW (Reuters) - Thousands of people gathered in Moscow Saturday to call for a disputed election to be rerun, increasing pressure on Vladimir Putin as he seeks a new term as Russian president.
The protesters were heartened before the second big rally in two weeks by the Kremlin's human rights council saying a new election should be held, although it is only an advisory body whose recommendations are regularly ignored by Russia's leaders.
Tens of thousands of people protested in Moscow on December 10 and many more demonstrated across the world's largest country the same day to complain against alleged vote-rigging in the December 4 election won by Putin's United Russia party.
Police said about 5,000 people attended the start of Saturday's rally on Prospekt Sakharova (Sakharov Avenue), named after Soviet-era dissident Andrei Sakharov. But a police source told Itar-Tass news agency the figure was around 20,000.
A large stage had been erected with a slogan above it declaring: "Russia will be free!."
Witnesses said about 50,000 attended the last rally in Moscow on December 10.
Many of the protesters carried balloons and flags and the atmosphere was relaxed. Police did not intervene at the last big Moscow rally, the largest opposition protest since Putin rose to power 12 years ago.
Bringing together liberals, nationalists, anarchists, environmentalists and urban youth, protest organizers settled on a list of 19 speakers that includes former Soviet President Mikhail Gorbachev, anti-corruption blogger Alexei Navalny, rock singer Yuri Shevchuk and writer Boris Akunin.
The opposition has rejected conciliatory efforts by Putin and President Dmitry Medvedev, saying they have ignored its key demand for a rerun of the December 4 parliamentary poll, which handed a slim majority to the ruling United Russia party.
The opposition says United Russia benefited from widespread voting irregularities and international monitors said the vote was slanted in the ruling party's favor.
The Kremlin human rights council said the alleged irregularities discredited the new State Duma lower house.
It said this posed "a real threat to the Russian state" and called for the resignation of Vladimir Churov, the head of the central election commission.
Medvedev promised Thursday to relax the Kremlin's grip on power, including measures to restore the election of regional governors and to allow half the seats in the State Duma lower house of parliament to be directly elected in the regions.
But opposition leaders said the moves were too little, too late by a leader who has carried out few of his reform promises since he succeeded Putin as president, who was barred by the constitution from a third successive term.
Police said they would allow protesters to bring thermos flasks, with weather forecasters predicting the coldest day of winter so far.
Gorbachev said he regretted supporting Putin when he rose to power and replaced President Boris Yeltsin more than a decade ago. "I feel ashamed," he told Novaya Gazeta liberal daily on Friday.
Putin's popularity has declined since he and Medvedev announced plans in September to swap jobs next year, a decision which many Russians said showed a disregard for democracy.
Putin, 59, has suggested that many of those taking part in the biggest protests since he came to power 12 years ago had been paid to turn out. He has accused the United States of stirring the protests and foreign powers for funding them - a theory that has spread across pro-Putin websites.
(Editing by Timothy Heritage)