A study by a company that helps track pirated digital books estimates that there were 9 million illegal downloads of copyrighted books in the final months of last year. Attributor, which works for publishers including Hachette Book Group and John Wiley & Sons, scanned 25 Web sites that offer readers downloadable content, looking for 913 titles across categories ranging from business and investing to fiction. It found, for example, that illegal copies of “Freakonomics,” by Steven D. Levitt and Stephen J. Dubner, were downloaded 1,082 times and “Angels & Demons,” by Dan Brown, 7,951 times. The study did not track any titles published by the company’s clients, which would exclude the oft-pirated “Twilight” series published by Hachette. Rich Pearson, general manager of Attributor, said although not every pirated copy represented a lost sale, the potential loss to the publishing industry could be as high as $3 billion. Some analysts doubted that piracy was as big a problem for the book industry as the study suggested. Mike Shatzkin, chief executive of Idea Logical Company, a book industry consultant, said many people who might illegally download an e-book would never have bought it in the first place.