Story highlights Protests are taking place in Moscow and other cities

Demonstrators think the parliamentary vote was rigged

Vladimir Putin's United Russia retained its majority

20,000 to 25,000 people turn out for protest in Russia, police say

Tens of thousands of Russians turned out in central Moscow and across the country Saturday to protest what they believe were rigged parliamentary elections.

United Russia, the party of Prime Minister Vladimir Putin, suffered big losses in the election, but retained its parliamentary majority. On Saturday, protesters chanted "Putin out," according to a correspondent from state-run RIA Novosti news agency.

Between 20,000 and 25,000 protesters had gathered in the capital, Moscow, Ria Novosti said Saturday, citing police. There have been no reports of unrest and security has been tight.

Vladimir Ryzhkov, co-chairman of the Party of People's Freedom, said 40,000 people had massed and some 10,000 were headed to the main protest venue, the news agency reported.

Rally organizers and Moscow city officials agreed to allow demonstrators on Bolotnaya Square in Moscow Saturday. The event has been authorized to hold 30,000 people, yet tens of thousands more have indicated that they want to attend.

JUST WATCHED Protests erupt in Russia over elections Replay More Videos ... MUST WATCH Protests erupt in Russia over elections 02:45

JUST WATCHED Nude protests, large rally in Moscow Replay More Videos ... MUST WATCH Nude protests, large rally in Moscow 02:17

JUST WATCHED Russian protests on social media sites Replay More Videos ... MUST WATCH Russian protests on social media sites 02:34

The protesters are demanding an annulment of the December 4 election results and a new vote.

Election officials on Friday released the official results: 238 seats for United Russia; 92 seats for the Communists; 64 seats for Fair Russia; and 56 seats for the Liberal Democrats. Protesters say that these results are fraudulent.

Putin said Russians have the right to assemble peacefully and legally.

Hundreds of protesters were arrested during demonstrations earlier this week. Police said they cracked down on those demonstrations because the turnouts were not legal and the protesters were being disorderly.

Demonstrators turned out in other parts of the country. RIA Novosti correspondents said around 1,000 people ralled in the southern Siberian city of Barnaul. Protests occurred in Krasnoyarsk, Vladivostok, Chita and Khabarovsk in eastern Russia.

A demonstration also was planned for St. Petersburg, Russia's second largest city.