(Adds comment, detail) LONDON, Feb 3 (Reuters) - The euro zone’s economic downturn deepened in January as renewed restrictions to quell the spread of the coronavirus hit the bloc’s dominant service industry hard, offsetting a robust performance by manufacturers, a survey showed on Wednesday. Coronavirus cases have been soaring again and governments across the continent have renewed tough lockdown measures, clamping down on public life and forcing hospitality and entertainment venues to close their doors. IHS Markit’s final January Composite Purchasing Managers’ Index (PMI), seen as a good guide to economic health, fell to 47.8 from December’s 49.1 but was a touch above a flash reading of 47.5. Anything below 50 indicates a contraction. “Today’s data support our view that the euro zone economy has started the new year on the back foot,” said Melanie Debono at Capital Economics. “With the vaccine rollout stalling already and new virus strains circulating, the risks to our forecasts for a recovery to start near the end of Q2 are firmly to the downside.” A PMI covering the service industry fell to 45.4 from December’s 46.4 but came in higher than the 45.0 flash reading. A manufacturing PMI on Monday showed factory growth remained robust at the start of the year but the pace waned. Business activity across the German service sector contracted for a fourth month running in January as a stricter lockdown hit businesses and put Europe’s largest economy on track for a first-quarter contraction. Meanwhile, French service sector activity slowed, albeit not as much as initially thought, weighing on broader private sector business. Italy and Spain also saw services activity shrink. In Britain, outside the European Union, a third national coronavirus lockdown has put the economy on course for a sharp contraction in early 2021, but services companies - buoyed by progress on vaccinations - were confident about recovery. “Their optimism is warranted, given that the UK’s vaccination programme is proceeding speedily,” said Samuel Tombs at Pantheon Macroeconomics. The euro zone economy contracted 0.7% last quarter, official data showed on Tuesday, and a January Reuters poll predicted it would take up to two years for the economy to return to pre-COVID-19 levels. With venues closed, demand for services unsurprisingly fell. The new business index dropped to 45.4 from 46.6, albeit ahead of the 44.7 flash estimate. But amid hopes the vaccines being rolled out would allow for some return to normality, overall optimism about the year ahead remained resilient. The composite future output index only nudged down to 64.2 from December’s 64.5, which was the highest since April 2018.