MOSCOW -- Vladimir Putin has never sent a tweet, but on his 59th birthday, the Russian prime minister became a Twitter sensation, with a hashtag devoted to his birthday rising near the top of global trends.

#CǏǀǑǈǁǎƯǓǒǈǍǓƧǀƽǒǎ -- which translates as Thanks to Putin for that -- was set up early Friday by Vladimir Burmatov, an activist from Mr. Putin's United Russia party, to gather rhyming praise for the party leader. But the topic was quickly taken up by Mr. Putin's critics -- who are numerous in Twitter's Russian-speaking...