(For a Reuters live blog on U.S., UK and European stock markets, click LIVE/ or type LIVE/ in a news window) Jan 19 (Reuters) - European shares rose on Tuesday after Asian markets rallied on optimism about China’s economy, while upbeat earnings reports from miner Rio Tinto and computer peripherals maker Logitech boosted sentiment. The pan-European STOXX 600 index gained 0.4% by 0809 GMT, while Germany’s DAX and France’s CAC 40 gained 0.4% and London’s blue-chip index rose 0.7%. UK-listed shares of Rio Tinto gained 1.4% after it reported a 2.4% rise in fourth-quarter iron ore shipments, helped by industrial activity in top consumer China. Miners were the top gainers among sectors, up almost 1%. Logitech jumped 6.9% as it raised its 2021 sales growth and profit outlook for the third time, benefiting from a pandemic-driven boost in demand for work-from-home products and gaming accessories. Asian stocks closed in on all-time highs as investors wagered China’s economic strength would help underpin growth in the region after data confirmed the world’s second-largest economy was one of the few to grow over 2020.