By Andrew Osborn and Maria Tsvetkova

MOSCOW (Reuters) - Russia told the United States on Thursday to get to the bottom of a hacking scandal involving Democratic Party emails itself and rejected what Donald Trump said was a sarcastic suggestion that Moscow should dig up Hillary Clinton's "missing" emails.

Trump, the Republican Party's presidential candidate, angered Democrats on Wednesday by inviting Russia to unearth tens of thousands of emails from rival Clinton's tenure as U.S. secretary of state.

He spoke out after President Barack Obama said it was possible Russia might try to influence the Nov. 8 U.S. presidential election after a leak of Democratic National Committee emails that experts blamed on Russian hackers.

Suggestions of Russian involvement have riled the Kremlin, which has categorically denied this and accused U.S. politicians of seeking to play on Cold War-style American fears of Moscow by fabricating stories for electoral purposes.

President Vladimir Putin has tried to avoid giving the impression of favoring any U.S. candidate, while hailing the populist Trump as being "very talented". Russian state TV coverage has tended to tilt towards Trump over Clinton.

On Thursday, Kremlin spokesman Dmitry Peskov said accusations of a Russian hand in hacking Democratic Party emails bordered on "total stupidity" and were motivated by anti-Russian sentiment. He rejected Trump's apparently sarcastic suggestion for Russia to dig up Clinton's emails point-blank.

"As regards these (email) batches, that is not our headache. We never poke our noses into others' affairs and we really don't like it when people try to poke their nose into ours," he said.

"The Americans need to get to the bottom of what these emails are themselves and find out what it's all about."

Trump, who has repeatedly said he would strive for better U.S. relations with Russia if he won the presidency, also raised eyebrows by saying he would consider recognizing Ukraine's Crimea as part of Russia, which annexed the region in 2014.

Peskov said the Kremlin was unmoved by that comment, and it would not change what he says has been its neutral stance on U.S. presidential candidates.

"We know perfectly well that candidates in the heat of a pre-election struggle say one thing, but that later, when under the weight of responsibility, their rhetoric becomes more balanced."

In response to Trump's remarks on Crimea, Ukraine's U.N. Ambassador Volodymyr Yelchenko said on Thursday: "Mr. Trump is not the president of the United States, at least not yet."

"Secondly, there are the well-known decisions of the United Nations. ... I'm pretty sure that any U.S. government will pay full respect to those decisions," he told reporters.

In March 2014, the 193-member United Nations General Assembly adopted a resolution declaring invalid Crimea's Moscow-backed referendum seceding from Ukraine.

Analysts say the Kremlin would welcome a Trump victory in November because the wealthy New York businessman has repeatedly praised Putin, spoken of wanting to get along with Russia, and has said he would consider an alliance with Moscow against Islamic State.

Trump's suggestion that he might abandon NATO's pledge to automatically defend all member states is also likely to have gone down well in Moscow, where the Western military alliance is cast as an outdated Cold War relic.

(Additional reporting by Lidia Kelly in Moscow and Michelle Nichols at the United Nations; Editing by Mark Heinrich and Jonathan Oatis)