* Obama appeals to Russian citizens on second day of trip
* Urges partnership to boost prosperity, security, democracy
* Muted reaction from Russian students, media
* Business leaders call for transparency, rule of law
(Adds meeting with Russian NGOs, church, Obama flying on to G8) By Jeff Mason and Michael Stott
MOSCOW, July 7 (Reuters) - President Barack Obama appealed to the Russian people on Tuesday to join the United States in overcoming past differences and building a prosperous democratic future free of corruption and the threat of nuclear war.
Broadening his message of a "reset" in relations between the two former Cold War superpowers on the second day of a visit to Moscow, Obama said citizens, business people and companies all had a part to play in improving ties and boosting trade.
"America wants a strong, peaceful and prosperous Russia," Obama told students at Moscow's New Economic School. "...Look to the future that can be built if we refuse to be burdened by the old obstacles and old suspicions".
Obama was careful in his speech, billed by the White House as a major setpiece, to avoid direct criticism of the Kremlin, where he agreed on Monday an arms-cutting package and permission for U.S. troops to cross Russia en route for Afghanistan.
Instead, Obama emphasised in his public appearances U.S. ideals such as prosperity, democracy and the rule of law, sending a more nuanced message.
"People everywhere should have the right to do business or get an education without paying a bribe," Obama said in his speech to students, making an oblique reference to Russia's ingrained culture of corruption.
The students, many sceptical of the United States after the chaos wrought by "wild East" capitalism in the post-Soviet period, listened politely to Obama in a hall near the Kremlin used mainly for official events but showed little enthusiasm.
Their reaction reflected a theme of Obama's visit -- a generally low-key reception by Russian media and ordinary citizens. Opinion polls show a majority of Russians mistrust the United States and believe it abuses its power.
NO OBAMAMANIA HERE
"We are maybe the one country in the world where there is no Obamamania," Sergei Markov, a parliamentary deputy from the ruling United Russia party, told Reuters.
"For us he is not president of the world but the president of the United States of America."
Later, at a summit with business leaders, Obama emphasised that a "reset" in prickly Russia-U.S. relations -- the main aim of his trip -- needed to involve many more people than himself and his Russian counterpart Dmitry Medvedev.
"This can't just be a matter for two presidents," he said. "It has to go deeper. It has to be between our people."
Trade between Russia and the United States amounted to just $36 billion last year -- about the same amount as Russian trade with Poland -- and investment is also relatively low.
Obama noted that the percentage share of Russian trade in total United States trade had not risen since the Cold War.
U.S. Commerce Secretary Gary Locke, who is visiting with Obama, said American business chiefs "emphasised the need for greater predictability, stability, transparency and the rule of law" during the trip.
Obama tried to mend fences with Russia's most powerful politician, Prime Minister Vladimir Putin, during a breakfast meeting at Putin's forest dacha which lasted around two hours..
The U.S. president had upset Putin before travelling to Moscow by describing him in an interview as a man with one foot planted in the old ways of doing things. But he went out of his way on Tuesday to praise Putin's "extraordinary work".
Putin, a former KGB spy who was president between 2000 and 2008 before handing over the Kremlin job to his hand-picked successor Medvedev, looked uncomfortable before the meeting but both sides said afterwards that it had gone well.
A senior U.S. administration official said Putin and Obama had a "very interesting and open discussion" which "formed the basis of a good relation upon which they can build", though they had disagreed on some issues.
As on some other foreign trips, Obama's remarks were modest for a U.S. leader, avoiding lecturing his audience or claiming he had all the answers.
"I think it's very important that I come before you with some humility," he told opposition leaders. "I think in the past there's been a tendency for the United States to lecture rather than to listen. And we obviously still have much work to do with our own democracy".
It was not clear, though, how much of Obama's message ordinary Russians would hear. His main speech to students was not broadcast live by major channels and state-controlled television mostly avoided conveying his words directly.
Not everyone was happy with Obama's conciliatory gestures towards the Russian leadership.
Roman Dobrokhotov, leader of the democratic youth movement "We", told Reuters by telephone from a police station in central Moscow that he was detained as he was walking to Obama's hotel with a protest poster.
"Russia's most topical, most burning issue at the moment is human rights and democracy. Our message to Obama is -- as long as there is dictatorship in Russia, you can't trust Medvedev."
Obama also met representatives of civil society in Russia, including the leader of the powerful Russian Orthodox Church, Patriarch Kirill. He will leave Russia on Wednesday morning for a G8 summit meeting in Italy. (Additional reporting by Matt Spetalnick, Oleg Shchedrov, Guy Faulconbridge, Dmitry Zhdannikov, Conor Humphries and Dmitry Solovyov; editing by Tim Pearce)