In his dynamite, accidentally topical new special “Miami Nights,” released for free on YouTube over the July Fourth weekend, Hannibal Buress describes himself as “medium famous,” which he says translates to: “I constantly talk people out of recognizing me.”

Comedy has traditionally been more difficult for those in the middle. A-list stars are given every break, and the press loves a Next Big Thing. But those in between face particular challenges these days with a glut of competition, and a Netflix approach that puts a premium on algorithmically informed celebrity. This helps explain the current booming DIY comedy-special movement that has accelerated this year with many veteran stand-ups, including Liz Miele, Mark Normand and Matt Ruby, releasing funny hours free online.

Some comics have taken this route in reaction to the constraints of establishment platforms. And in an introduction, Buress appears to echo this critique when he says he self-produced “Miami Nights” because YouTube provided him “spontaneity and freedom that some other outlets wouldn’t.”

There is surely truth in this. At a time when specials filmed last year appear to belong to a different era, Buress shoehorned a COVID-19 joke into the credits if you freeze-frame and look carefully. But almost as soon as he made the point, Buress undercut it, saying: “Imma be real: This is an ad.” Then he shouted out his sponsor.

Buress is not one to get on a soapbox. What makes him such an unusual comedic voice is that he has built his distinctive sensibility on a quality rare among stand-ups: nonchalance. Buress talks about anxiety-provoking subjects like dying, asthma attacks and run-ins with the police with casual ease, countering the tension of danger with a deep well of silliness.

“Miami Nights” is not the lo-fi production you expect from YouTube. It is actually his slickest special yet, complete with jokes that use Auto-Tune, multimedia, some “Miami Vice”-era retro visuals and dreamy camerawork. Kristian Mercado’s playfully flashy direction warms up the crowd with bursts of neon and hip-hop, and often positions the camera at a mouse’s-eye view. It is amusingly apt that the first special that films Buress in a way to make him look like a glamorous star is on a site known for homemade cat videos.

Buress has the kind of gifts that seemed destined for superstardom, but he never exactly got there, which is curious only if you think fame is distributed fairly. His particular level of success is a recurring theme in this hour. The special begins with a shot of him doing jokes as a teenager, and then it pivots into an opening bit in which he says he gets asked to host game shows twice a year. His response, in a computer-generated demonic bass voice, is: “The prophecy will be fulfilled but the time is not now.”

Buress has become a go-to performer for supporting roles in offbeat shows like “Broad City” or “The Eric Andre Show” as well as Hollywood comedies like “Tag,” but not the focus of his own vehicle, beyond comedy specials. His most viral moment might have come when he responded to Bill Cosby’s scolding of young people by reminding the audience of rape allegations against him, a comment made before prosecutors looked again at the accusations. Underneath his nonchalance, Buress can be a brutal counterpuncher.

He shows that off again in the standout section of this special when he recounts the story of his arrest in Miami for disorderly intoxication. (Early on, he says he is now sober.) For more than 20 minutes, he describes a confrontation with a police officer that gets hostile after he jokingly asks the officer to get him an Uber. It eventually leads to the officer arresting Buress after he spoke to the officer’s body camera as opposed to his face, which Buress explains as a result of his career in show business. “I’m a professional,” he says with mock umbrage.

From his cell at the police station, he continues to roast the officer and then asks to discuss the mug shot as if it were a photo shoot. “As I always do, I request a preproduction meeting with the photographer to discuss vision,” he says. It’s a riotously funny story that takes an abrupt turn at the end when he finds out that the officer was arrested himself after choking someone in a bar, then fleeing the scene. “How about if you run away from the cops, you can’t be a cop anymore?” Buress says. “Why is this dude still working?”

Buress, who filmed this special in August 2019, keeps the focus of his story narrow and the way he spins it is far less overtly political than many other comics’ bits criticizing the police. But the meaning of great comedy adjusts to the moment, and by releasing it now, with the Black Lives Matter movement at the center of public consciousness, Buress makes his own statement. Even though he makes the arrest sound like a benign adventure, he is describing overly aggressive policing that evades accountability and escalates a minor transgression into something far worse. He chuckles throughout, often at himself, but there is a darkness here that will be more visible to some people now than when he shot it.

Buress never gets particularly grave, but he bristles under the surface. His story plays like a comic revenge fantasy, one that stings because it is real, something he keeps reminding his audience. Buress doesn’t just name the police officer. He shows local news coverage and quotes news articles. He even introduces his defense lawyer in the audience. He explains that after he was arrested, he found three suitable options but went with this guy, because his last name was Bieber, because, hey, why not?