LONDON—The eurozone’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 Feb. 3. 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 [Feb. 3] data support our view that the eurozone 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 …