Apple Music has garnered some 15 million users in its three short months in business, music industry sources told The Post.

The figure is said to be a home run for Apple Music execs, whose projected number was on par, but who conceded that a slow first quarter ramp-up would be the norm, sources said.

Around half of those who signed up have not turned off auto-pay, which means they’ll convert from free to paid customers, sources said. The initial free three-month trial ends on Sept. 30.

If those 7.5 million Apple subscribers stay with the service after the gratis period, the company will have more than a third of Spotify’s 20 million paid subscribers. Spotify launched globally eight years ago and in the US in July 2011.

Spotify has 75 million users, with the majority not paying for its premium tier.

“If Apple reports between 30 million and 50 million trialists by year-end, then we can consider it successful,” music industry analyst Mark Mulligan of MIDiA Research told The Post.

“More than 5 million paying subscribers would be a success. Above 7 million would be a strong success,” he added.

Mulligan also said that even a 25 percent conversion rate of free to paid would be positive. He predicts that 35 to 40 percent of users might make the first payment (due Oct. 1) and then 20 to 25 percent would make a second payment.

Back in early August, Eddy Cue, Apple’s senior vice president of Internet software, said the service attracted 11 million sign-ups. “That was a success,” said Mulligan.

Apple Music may have some other catalysts around the corner to help boost its numbers.

The launch of iPhone 6s, which is said to come pre-loaded with the Apple Music app, is expected to make it even easier to give it a try. The iOS 9 update also fixed the bugs that plagued the service at its outset. The new software also enables push notifications.

Apple Music is also spending millions on TV ads just ahead of the fourth quarter.

Its first big splashy campaign came during the MTV VMAs and the latest during Sunday night’s Emmy Awards on Fox. The MTV ad featured singer The Weeknd, who shot to No. 1 on the Billboard Hot 100 with “Can’t Feel My Face.”