Darrell Jackson, proceeding pro se, appeals the district court’s dismissal of his third amended complaint, in which he brought claims under 42 U.S.C. § 1983 for alleged constitutional violations stemming from his 2008 arrest in Orange County, Florida.
I.
Jackson first contends that the district court erred when it dismissed his claims against Keith Vidler and Eric Capraun, two officers involved in his arrest, and Kevin Beary, the sheriff of Orange County at the time he was arrested, based on qualified immunity. We “review de novo a district court’s order dismissing a complaint, accepting all allegations in the complaint as true and construing the facts in a light favorable to the plaintiff.” Fortner v. Thomas, 983 F.2d 1024, 1027 (11th Cir.1993). A district court may dismiss a complaint for failure to state a claim upon which relief can be granted when the complaint’s allegations “indicate the existence of an affirmative defense, so long as the defense clearly appears on the face of the complaint.” Id. at 1028.
Qualified immunity protects government officials acting within their discretionary authority “unless their conduct violates ‘clearly established statutory or constitutional rights of which a reasonable person would have known.’ ” Andujar v. Rodriguez, 486 F.3d 1199, 1202 (11th Cir.2007). Once the defendant has established that he was acting within his discretionary authority, “the burden shifts to [the plaintiff] to establish that the defendants’ conduct violated clearly established law.” Harbert Int’l v. James, 157 F.3d 1271, 1284 (11th Cir.1998).
A.
Jackson contends that Capraun and Vidler violated his Fourth Amendment rights by falsely arresting him. According to Jackson’s complaint, he was arrested as part of a sting set up by the Orange County Sheriffs Department to catch bicycle thieves. Vidler placed a bike in the *856street and the officers watched it. Unaware of the watching officers, Jackson picked up the bike and rode away on it. Capraun monitored Jackson as he rode away, and another officer1 arrested him. After Jackson’s arrest, Capraun transported him to the booking center.
The allegations indicate that in placing the bicycle on the street and arresting Jackson, Vidler and Capraun were participating in a sting operation that was “undertaken pursuant to the performance of’ their duties as police officers and was therefore within the scope of their authority. See Harbert Int’l, 157 F.3d at 1282. The burden then shifts to Jackson to show that they violated his constitutional rights.
“A warrantless arrest without probable cause violates the Fourth Amendment and forms a basis for a section 1983 claim.” Rodriguez v. Farrell, 280 F.3d 1341, 1345 (11th Cir.2002). “Probable cause to arrest exists when law enforcement officials have facts and circumstances within their knowledge sufficient to warrant a reasonable belief that the suspect had committed or was committing a crime.” United States v. Floyd, 281 F.3d 1346, 1348 (11th Cir.2002). “In the context of a claim for false arrest, an officer is entitled to qualified immunity where that officer had arguable probable cause, that is, where reasonable officers in the same circumstances and possessing the same knowledge as the Defendants could have believed that probable cause existed to arrest the plaintiff.” Davis v. Williams, 451 F.3d 759, 763 (11th Cir.2006) (quotation marks omitted).
Under Florida law, a defendant commits theft if he:
knowingly obtains or uses ... the property of another with intent to, either temporarily or permanently:
(a) Deprive the other person of a right to the property or a benefit from the property.
(b) Appropriate the property to his or her own use or to the use of any person not entitled to the use of the property.
Fla. Stat. § 812.014(1). Because the officers saw Jackson appropriate a bicycle that he knew was not his, they had arguable probable cause to arrest him for theft. Jackson argues that he believed the bike had been abandoned, but a reasonable officer still could have concluded that Jackson took the bike with the intent to appropriate the property of another, in violation of Fla. Stat. § 812.014(1).2 And there is no clearly established law that indicates that a bicycle theft sting, like the one used by these officers, is a violation of constitutional rights. Both Vidler and Capraun are entitled to qualified immunity for Jackson’s false arrest claims.
B.
Jackson also contends that Ca-praun violated his Fourteenth Amendment rights by showing a deliberate indifference to his medical needs.3 According to Jackson’s complaint, he was suffering from acute back pain, which had been caused by *857another officer tackling him off of the bike to arrest him. He told Capraun that he needed medical care for his acute back pain, but Capraun said that Jackson would have to wait until he was through with his report. Twice while Capraun was preparing the report, he stopped to assist in other arrests involving the bicycle sting operation. As part of assisting in the arrests, Capraun sped down bumpy dirt roads with Jackson riding in the back seat of the patrol car, exacerbating Jackson’s back injury. Capraun eventually took Jackson to the Orange County Booking and Release Center, where he was seen by medical staff and treated for his injuries upon arrival.
To prevail on a claim of deliberate indifference to serious medical need, a plaintiff must show: “(1) a serious medical need; (2) the defendant’s deliberate indifference to that need; and (3) causation between that indifference and the plaintiffs injury.” Youmans v. Gagnon, 626 F.3d 557, 563 (11th Cir.2010) (alterations and quotation marks omitted). “A serious medical need is one that has been diagnosed by a physician as mandating treatment or one that is so obvious that even a lay person would easily recognize the necessity for a doctor’s attention.” Id. at 564 (quotation marks omitted). Although Jackson alleges that he told Capraun about his back pain, he does not allege that his pain had been diagnosed as mandating treatment and or that it would have been obvious to a layperson that he needed treatment, so he has not sufficiently alleged a “serious medical need.” See id. at 561 (concluding that the plaintiff had not established serious medical need when he had “several cuts and abrasions on his head, face, shoulder, elbow, and hand” and appeared to be bleeding slightly).
And even if we assume that Jackson’s back pain did constitute a “serious medical need,” Capraun’s actions, as alleged in Jackson’s compliant, do not rise to the level of a deliberate indifference. “To prove ‘deliberate indifference’ to a serious medical need, a plaintiff must show (1) subjective knowledge of a risk of serious harm; (2) disregard of that risk; (3) by conduct that is more than gross negligence.” Id. at 564 (alterations and quotation marks omitted). Jackson alleged that Capraun delayed his treatment long enough for Capraun to fill out the arrest report and to help in the arrest of two other people. That kind of delay does not demonstrate that Capraun disregarded the risk to Jackson “by conduct that is more than gross negligence.” See id. at 566 (concluding that the booking officer did not violate the plaintiffs constitutional rights when he spent three hours interviewing and booking the plaintiff before getting him treatment for visible cuts and bruises caused by the arresting officers).
C.
Jackson also contends that Kevin Beary, who was the sheriff at the time Jackson was arrested, violated his constitutional rights under a respondeat superior theory of liability and because he designed the bike sting operation.
“It is well established in this Circuit that supervisory officials are not liable under § 1983 for the unconstitutional acts of their subordinates on the basis of respon-deat superior or vicarious liability.” Cottone v. Jenne, 326 F.3d 1352, 1360 (11th Cir.2003). But a supervisor may be liable under § 1983 when his “custom or policy ... result[s] in deliberate indifference to constitutional rights.” Rivas v. Freeman, 940 F.2d 1491, 1495 (11th Cir.1991). Because Jackson failed to allege sufficient facts to show that the officers’ actions in executing the bike theft sting violated his constitutional rights, he failed to show that *858any such custom or policy instituted by Beary in connection with the sting operation resulted in a deliberate indifference to his constitutional rights. The district court did not err in granting Vidler, Ca-praun, and Beary’s motions to dismiss based on qualified immunity.
II.
The district court dismissed sua sponte Jackson’s allegations against the officer who participated in the arrest with Capraun and Vidler because Jackson failed to serve him. “[W]e review for abuse of discretion a court’s dismissal without prejudice of a plaintiffs complaint for failure to timely serve a defendant under Rule 4(m).” Rance v. Rocksolid Granit USA, Inc., 583 F.3d 1284, 1286 (11th Cir.2009). A district court’s decision will not be disturbed on abuse of discretion review if it falls within a range of permissible choices and is not influenced by a mistake of law. Zocatas v. Castro, 465 F.3d 479, 483 (11th Cir.2006).
Jackson did not know the name of this officer but alleged that “Officer Doe” was the officer who physically took him into custody, and, in doing so, used excessive force. Jackson alleged that Officer Doe pushed him backwards off the bicycle, causing him to fall on the ground and injure his back. Doe then dove on top of him, even though Jackson asserts that he never resisted or attempted to evade arrest. The court dismissed this claim because Jackson failed to identify and then serve the officer within the time limit set by the court.
Under Federal Rule of Civil Procedure 4(m),
If a defendant is not served within 120 days after the complaint is filed, the court — on motion or on its own after notice to the plaintiff — -must dismiss the action without prejudice against that defendant or order that service be made within a specified time. But if the plaintiff shows good cause for the failure, the court must extend the time for service for an appropriate period.
“Good cause exists only when some outside factor, such as reliance on faulty advice, rather than inadvertence or negligence, prevented service.” Lepone-Dempsey v. Carroll County Com’rs, 476 F.3d 1277, 1281 (11th Cir.2007) (quotation marks and alterations omitted).
Jackson argues that he had good cause for the failure to serve the officer because he could not identify Doe. The district court gave Jackson 45 additional days to successfully effect service on Doe and warned Jackson that a failure to timely complete service would result in a dismissal of the action without further notice. To facilitate service, the court also provided Jackson with the requisite forms, as well as a list of instructions for their completion. Because the district court instructed Jackson on the proper procedure for identifying and serving Doe, extended his filing deadlines, and warned him that a failure to comply would result in dismissal, and Jackson still failed to identify and serve Doe, the court acted within its discretion by dismissing his claim without prejudice for failure to serve under Rule 4(m).
III.
After Jackson’s arrest, he was charged with grand theft and possession of drug paraphernalia4 and prosecuted by Antonio Tapia, the assistant state attorney. Jack*859son brought a claim against Tapia for malicious prosecution, and the district court dismissed that claim based on prosecutorial immunity. “Prosecutors performing ‘prosecutorial functions’ receive absolute immunity and are therefore not subject to suit under 42 U.S.C § 1988.” Long v. Satz, 181 F.3d 1275, 1278 (11th Cir.1999). If a defendant asserts prosecutorial immunity in a Rule 12(b)(6) motion, “we ask if the allegations of the complaint disclose activities protected by absolute immunity.” Id. at 1279. If the answer is yes, the defendant is immune from suit. Id. “Pros-ecutorial immunity applies ... to the prosecutor’s actions in initiating a prosecution and presenting the State’s case.” Hart v. Hodges, 587 F.3d 1288, 1295 (11th Cir.2009).
Jackson’s third amended complaint alleged that Tapia initiated the prosecution against him with malicious intent, which Tapia displayed by calling Jackson a “crack head” in an email to Jackson’s criminal defense attorney. Tapia is entitled to absolute immunity for initiating the prosecution even if he did it with malicious intent. See Hart, 587 F.3d at 1295 (“A prosecutor is immune for malicious prosecution.”). The district court did not err in dismissing Jackson’s claim against Tapia based on prosecutorial immunity.
IV.
Finally, Jackson argues that the court erred in sua sponte dismissing his claim against Orange County under 28 U.S.C. § 1915(e)(2) for failure to state a claim upon which relief can be granted. We review that dismissal de novo, viewing the allegations in the complaint as true. Hughes v. Lott, 350 F.3d 1157, 1159-60 (11th Cir.2003).
Jackson alleged that since 1981, he had been arrested five times by the Orange County Sheriffs Department. He asserted that he has also witnessed the county deputy sheriffs entrapping citizens numerous times using illegal sting operations. He contends that the illegality of those operations and the policies supporting them was so obvious that the County had at least constructive knowledge of the widespread abuse but failed to take any remedial steps to stop it.
Under Florida law, counties are not protected from suit by sovereign immunity. Abusaid v. Hillsborough Cty. Bd. of County Comm’rs, 405 F.3d 1298, 1314 (11th Cir.2005). However, “[w]hen an officer is sued under Section 1983 in his or her official capacity, the suit is simply another way of pleading an action against an entity of which an officer is an agent.” Id. at 1302 n. 3. In this case, that means that Jackson’s suit against Beary was essentially another way of pleading an action against Orange County. As we have discussed, Jackson’s complaint against Beary was properly dismissed; his complaint against Orange County was properly dismissed for the same reasons. To the extent that Jackson alleges claims against Orange County that he did not allege against Beary, they are based on allegations that are too vague to state a claim upon which relief may be granted.
AFFIRMED.