Thousands take to Moscow streets, calling for fair elections

• NEW: Thousands brave the bitter cold in Moscow to call for fair elections
• The mass protest follows an earlier one this month after parliamentary election results were announced
• Protesters want an investigation into this month's election results

Moscow (CNN) -- Thousands of people took to the streets Saturday in Moscow, braving bitterly cold weather to demand for fair elections after what they claim were rigged results earlier this month that returned Vladimir Putin's party to power.

The protest, organized primarily through social media and word of mouth, comes on the heels of an announcement by President Dmitry Medvedev of sweeping political reforms, an effort to address discontent following the December 4 parliamentary elections.

The latest mass protest follows one earlier this month, when tens of thousands of people across Russia turned out to protest the election results that kept Putin's ruling United Russia party in power, albeit with a smaller majority. Police estimated crowds in Moscow at 25,000, while organizers said at least twice as many participated.

The protests were considered -- among analysts and political observers -- the largest in Russia in the past two decades.

More than 40,000 people were expected to turn out Saturday, according to a Facebook forum discussion moderated by organizers.

Speaking this week before the newly elected parliament members in the Kremlin's St. George Hall, Medvedev proposed that Russia return to direct elections of regional governors; simplify the registration of political parties and presidential candidates; and establish a new editorially-independent national public TV channel.

Medvedev also called for lifting many of the political restrictions imposed in the past several years by his predecessor, Putin, Russia's current prime minister and candidate for the March 2012 presidential elections.

He also announced a number of new anti-corruption measures and called the redistribution of power and financial resources from the federal government to local governments across the country.

At the same time, he rejected widespread public criticism of the parliamentary elections, which were reportedly marred by fraud and other irregularities, and blamed anti-Kremlin opposition figures for their "attempts to manipulate the people and foment social discord."

"We will not allow instigators and extremists to involve society in their reckless schemes, nor will we tolerate interference in our internal affairs from the outside," Medvedev said.

"Russia needs democracy, not chaos. We need to have a faith for the future and justice. It is a good sign that society is changing, and citizens are expressing their position more actively, setting legitimate demands to the authorities. It is a sign that our democracy is growing more mature."

Protest organizers say Medvedev, who announced the reforms during his fourth and final state-of-the-nation speech Thursday, said he failed to address what authorities are planning to do about the recent alleged voting fraud, as well as whether fair and free elections are guaranteed in the future.