COFFEY, Circuit Judge.
On July 15, 1997, the defendant-appellant Robert Amerson (“Amerson”) was convicted before a jury of possession of cocaine base with intent to distribute in violation of 21 U.S.C. § 841(a)(1). On appeal, Amerson argues that the trial court committed error: 1.) in refusing to allow the introduction of the affidavit of Timothy Heard; 2.) in refusing to find that the prosecutor improperly distorted the burden of proof during closing argument; 3.) in denying Amerson’s request for a Franks hearing; and 4.) in refusing to hold that the evidence was insufficient to support a conviction. We affirm.
I. BACKGROUND
At approximately 7:30 p.m. on February 11, 1997, officers of the City of Peoria, Illinois, Police Department executed a search warrant at an upstairs unit of a two-story, multi-unit apartment building located at 1418 N.E. Adams Street, Peoria, Illinois. Three police officers, Officers Mumaw, Mitchell, and Anderson, approached the second floor unit by ascending a flight of stairs leading to a landing area just outside the apartment’s kitchen door. The landing area outside the kitchen door was monitored with a “sensor light,” and as the officers ascended the stairs and approached the door, the light activated. Officer Mumaw knocked on the door and announced loudly “Police, search warrant!”, and at this time, the apartment’s lessee, Jesse Tolliver (“Tolliver”), appeared at a window next to the kitchen door and shouted “It’s the police!” and disappeared from view. At the same time, the officers heard the sound of people running inside the apartment, and they forced open the door and entered the apartment.
Following a plan established prior to the raid, Officer Mumaw ran into the kitchen, down a hallway, and into a bedroom located in the rear corner of the apartment. At trial, Mumaw testified that when he arrived in the bedroom, he observed a black male identified as Timothy Heard (“Heard”), along with two black females, Sharon Parker and Angela Hill. He saw Heard run to the bedroom window, which was covered with a large, heavy curtain. According to Mumaw, it appeared that Heard “tr[ied] to punch a small 35 millimeter canister1 out the window, but it didn’t make it because the curtain didn’t go out the window....” As Heard was punching [679]*679at the curtain and trying to dispose of the film canister, Officer Mumaw tackled him. A second officer, Mitchell, entered the room and assisted Mumaw in placing Heard in custody, and at this time, Mu-maw observed the canister lying on the floor inside the bedroom near the window adjacent to the curtain. Inventory and analysis of the contents of the film canister revealed seven individually wrapped, small rocks of cocaine weighing approximately 2.7 grams total.
At the time Heard struck and broke the glass in the window through the heavy curtain in the second-floor bedroom, three other officers were stationed outside and were standing directly below the window, Officers Moore, Kirwan, and Patterson. Officer Patterson “heard glass breaking from above [him]. At that time [he] looked up and saw pieces of glass falling down just in front of [him].” Officers Moore, Kirwan, and Patterson immediately searched the ground with their flashlights and found nothing but pieces of a broken window glass. The three officers then proceeded up the stairs in order to help the officers inside with the search of the apartment.
After arresting Heard and securing the other occupants of the apartment, Officer Mitchell, the lead officer, stationed himself at the front door and proceeded to assist the officers in the conducting of a “sting” operation upon those individuals who they anticipated might be appearing at the front door to sell or purchase drugs. Between 7:30 p.m. and 8:30 p.m. three people came to the door. The first group, two black females, arrived and asked for Tol-liver. Officer Mitchell refused to allow them entry, and they left. The other individual was a white female, who was arrested by Officer Mitchell after attempting to purchase a “look-alike”2 drug for $50.
At 8:50 p.m., approximately one hour and twenty minutes after the warrant entry into the apartment, another officer, Officer Couve, assigned to maintain surveillance and positioned across the street from the apartment, radioed the officers in the apartment that a black male was approaching the apartment on foot. The officers inside the apartment waited until they heard a knock, whereupon Officer Mitchell queried the person knocking, “Who is it?” The person (Amerson) outside the kitchen door responded “It’s me, Rob,” at which time Tolliver, under arrest and inside the apartment, shouted “Rob, it’s the police!”, alerting the officers to open the door. Officer Mitchell testified that when he opened the door he saw Amerson, illuminated by the sensor light, standing on the porch just three feet away, and holding a plastic bag in his hands that contained a white substance. According to Officer Mitchell, upon observing the officers, Amerson immediately “thr[ew] his hands in the air above his head, at which time this object [the plastic bag] came out of his hand and went over his head, off the landing and onto the ground behind him.” Officers Moore, Mumaw, and Mitchell “reach[ed] out and grab[bed] Mr. Amerson and ... pull[ed] him inside the apartment and after a struggle he [was] handcuffed” and arrested. Officer Moore then went “downstairs and ... look[ed] outside on the grass for what [he] believed at the time to be cocaine thrown from [Amer-son’s] hands and that’s what [he] found laying on the ground.” Officer Moore located the cocaine “fairly quickly” in a location he described as follows: “if you just walked out of the kitchen and onto the landing area and looked straight down over the railing, you could see the object in the grass directly in front of the porch.” Officer Moore, who was using his flashlight to search, discovered a plastic bag containing a white substance, which the police seized and was later analyzed and found to be 6.4 grams of crack cocaine.
The defendant Amerson was arrested and charged with possession of cocaine base with intent to distribute in violation of [680]*68021 U.S.C. § 841(a)(1). Before trial, the defendant filed a motion with the court and requested a Franks hearing, arguing that the investigating officer’s affidavit in support of a search warrant was based on the false statement of an informant who had claimed to have seen Amerson at the apartment on a prior occasion during a time frame Amerson denied being at the apartment. The court denied the motion, holding that Amerson failed to establish that the investigating officer’s affidavit was false and ordered the parties to proceed to trial. At trial, Amerson called Heard to the witness stand, and Heard invoked his Fifth Amendment privilege and refused to testify. In response, the defense counsel proffered Heard’s signed affidavit, which stated:
I Timothy Herd [sic] do solemly [sic] swear that the statement I’m about to sign is correct and is given at [sic] my own free will. On February 11th 1997 between 6:30-7:30 p.m. at 1418 N.E. Adams I heard police enter the apartment. Being in possession of two packages of cocaine. I ran to the back bedroom and attempted to dispose of them. One landed inside. Later one was found outside. Robert Amerson was unware [sic] of any of this, for he was not yet present, but was later charged.
The government objected to the admission of the affidavit, contending that it was hearsay under Fed.R.Evid. 802 since it was made out of court and was offered for the truth of the matter asserted. The defense counsel responded that the document was admissible hearsay under Fed.R.Evid. 804(b)(3) because it was a statement against Heard’s penal interest. The judge refused to accept the affidavit, ruling that Rule 804(b)(3) requires that the affidavit be supported by corroborating circumstances demonstrating the truth of the out of court assertion and that Amer-son had failed to demonstrate the existence of corroborating circumstances.
The defense also objected to the prosecutor’s statement during closing argument: “[w]hat you’ve got here are vice and narcotics officers — experienced drug enforcement officers whose job it is to arrest dope peddlers ... ”, and “[t]hose officers all indicated that it was part of their duties to rid our streets of cocaine.... ” Amerson’s counsel argued that the prosecutor’s statement amounted to vouching for the police officer witnesses. The judge sustained the defendant’s objection. Later in the closing argument, the prosecutor stated: “[b]ot-tom line here, ladies and gentlemen, is this. Who is telling the truth and who is lying here.... You simply cannot believe the testimony of these police officers and believe the defendant’s testimony at the same time.” Amerson’s counsel once again objected, stating that the prosecutor was distorting the burden of proof. The trial judge overruled this objection and found that the prosecutor’s statements were appropriate since “there was direct testimony from each side” regarding whether Amerson threw the plastic bag of cocaine.3
At sentencing, the trial judge determined that Amerson was a career offender4 (pursuant to U.S.S.G. § 4B1.1) and had a guideline sentencing range of 360 months to life imprisonment. Nevertheless, the judge departed downward because Amerson, who was 52 at the time of sentencing, had a serious heart condition with a life expectancy of less than 360 months. The judge stated: “taking these two items together, that is your medical condition and the belief that I have that the substance of your prior convictions do [681]*681not justify what in effect would be a life sentence, it is my decision to depart downward from the guidelines by cutting your sentence in half.” On December 5, 1997, the judge imposed a sentence of 180 months imprisonment followed by an eight year term of supervised release.
II. ISSUES
On appeal, Amerson raises four issues, arguing that the trial court committed error: 1.) in refusing to allow the introduction of the affidavit of Timothy Heard; 2.) in refusing to find that the prosecutor improperly distorted the burden of proof during closing argument; 3.) in denying Amerson’s request for a Franks hearing; and 4.) in refusing to hold that the evidence was insufficient to support a conviction.
III. DISCUSSION
A. The Heard Affidavit
At trial Amerson requested that Heard’s affidavit be admitted as a declaration against penal interest under Federal Rule of Evidence 804(b)(3).5 The trial court refused to admit the statement, ruling that Amerson had failed to demonstrate that there existed “corroborating circumstances which clearly indicate the trustworthiness of the statement. I can not in good conscience find that that ... element has been met.”
We begin by noting that “out-of-court statements are generally inadmissible because they are presumed to be unreliable.” United States v. Hall, 165 F.3d 1095, 1110 (7th Cir.1999) (quoting United States v. Hooks, 848 F.2d 785, 796 (7th Cir.1988)). “We review the district court’s decision that a particular hearsay statement is admissible under Rule 804 only for an abuse of discretion.” United States v. Doerr, 886 F.2d 944, 954 (7th Cir.1989) (citation omitted). “Under the three part test we set forth in United States v. Garcia, 897 F.2d 1413, 1420 (7th Cir.1990), for determining the admissibility of a statement under Rule 804(b)(3), a court must find that, (1) the declarant’s statement was against the penal interest of the declarant, (2) corroborating circumstances exist indicating the trustworthiness of the statement, and (3) the declarant must be unavailable.” United States v. Moore, 936 F.2d 1508, 1516 (7th Cir.1991) (internal quotations omitted). Further, we have previously held that we will not
second-guess the sentencing judge[’s determinations] because he or she has had the best opportunity to observe the verbal and nonverbal behavior of the witnesses focusing on the subject’s reactions and responses to the interrogatories, their facial expressions, attitudes, tone of voice, eye contact, posture and body movements, as well as confused or nervous speech patterns in contrast with merely looking at the cold pages of an appellate record.
United States v. Garcia, 66 F.3d 851, 856 (7th Cir.1995) (citation, internal quotations, and emphasis omitted).
Heard’s affidavit states: “Being in possession of two packages of cocaine. I ran to the back bedroom and attempted to dispose of them. One landed inside. Later one was found outside.” (emphasis add[682]*682ed). Heard’s affidavit is untruthful for several reasons. Initially, Heard contends that he was in possession of two packages of cocaine and that “one landed inside” while the other “was found outside.” Though Heard refers to two packages of cocaine in his affidavit, he never affirmatively states that the second package of cocaine “found outside” just below the front porch and not in the vicinity of the bedroom window was the same package he attempted to dispose of outside the bedroom window. Through the mechanism of a carefully drafted affidavit, Heard attempts to deceive this Court into believing that the second package, found far and away from the bedroom window in question (thirty to forty feet to the east) and just below the front porch, belonged to him (Heard), not Amerson, and thus was not the bag of cocaine thrown by Amerson over his head and off the front porch and found immediately by Officer Moore. All this despite the fact that Officer Mumaw testified that Heard was prevented from propelling anything out of the window, for “[tjhere was a large curtain in the way where this gentleman was trying to punch” and the canister found on the floor, containing seven individually wrapped packages of crack, “didn’t make it because the curtain didn’t go out the window.” As Officer Patterson testified, he, along with Officers Moore and Kirwan, was standing directly and immediately below the broken bedroom window and “heard glass breaking from above [him]. At that time [he] looked up and saw pieces of glass falling down just in front of [him].” The three officers immediately searched the surrounding area with their flashlights and found nothing but broken glass. An hour later, Amerson appeared on the scene (at the front porch). Three officers (Officers Moore, Mitchell, and Mumaw) arrested him after observing a bag of cocaine in his hands and observing him (Amerson) throw the plastic bag of rock-like crack cocaine “over his head, off the landing and onto the ground behind him.” At this time, Officer Moore immediately descended the front steps and retrieved Amerson’s plastic bag containing crack cocaine. The affidavit states that Heard “attempted to dispose of’ both packages. Again, Heard, through craftily framed phrases, does not affirmatively state that he disposed of the second package, only that he “attempted to” dispose of the package.
In addition, the location where the police officers discovered the crack cocaine (immediately below the porch where Amerson was standing) makes that part of Heard’s affidavit wherein he states that he “attempted to dispose” of the two packages of cocaine and “[l]ater one was found outside” implausible. According to the evidence at trial, the bag of cocaine found outside the building immediately below the porch where Amerson was standing was found to be “at least thirty to forty feet” east and slightly to the south of the broken window and eight to ten feet south and west of the porch (which, like the bedroom window, was on the south side of the building). It is impossible to comprehend how Heard, who dropped his one canister of cocaine in the bedroom at the time he was tackled by Officer Mumaw, would have been able to project the second package out the window and in a direction east and slightly to the south of the broken window and parallel to the building through a heavy curtain a distance of thirty to forty feet while he was in the process of falling down as he was being tackled and restrained by two officers. We are of the opinion that the trial court’s exclusion of the affidavit was proper, for the information contained therein, that Heard attempted to dispose of a second package of crack through a scenario of action physically untenable without the guidance system of the latest “smart” bomb, was not only less than credible, but comes within the realm of impossibility.
In response, Amerson contends that the fact that Heard broke the window corroborates that particular aspect of the affidavit’s truthfulness. Granted both parties do agree that Heard broke the window, but they disagree as to whether Heard was [683]*683able to dispose of the second package of cocaine out the bedroom window, for the Government contends that Heard never had possession of another container of crack cocaine. Thus, the Government argues that Heard could not dispose of something he never had possession of. We fail to understand how Amerson’s contention that Heard broke the window corroborates Amerson’s version of events, for it merely corroborates that he broke the window, as the Government concludes. Furthermore, Amerson argues that the testimony of Sharon Parker, also arrested with Heard at the time of the police search, corroborates Heard’s affidavit. Without being specific, Parker merely testified that Heard told her some time after he was in custody that he had “several” packages of cocaine6 in his possession at the time of the search warrant entry.7 Though it is true that Heard had a single canister which contained seven individually wrapped rocks of cocaine, Heard’s affidavit is not corroborated by Parker’s testimony because his affidavit stated that he had but “two packages” of cocaine, not “several” as Parker stated under oath.8 Additionally, Parker’s testimony fails to even address the question of whether Amerson was carrying crack when he arrived on the porch at the apartment and was confronted by the three police officers.
Because both Parker’s testimony and Amerson’s contention that Heard disposed of the second package of cocaine are inconclusive, there is no evidence in the record to corroborate Heard’s ambiguous and untruthful affidavit. Thus, the trial judge did not err in exercising his “considerable discretion,” United States v. Nagib, 56 F.3d 798, 805 (7th Cir.1995), when he refused to receive the affidavit into evidence. See United States v. Bradley, 145 F.3d 889, 892 (7th Cir.1998) (A court “abuse[s] its discretion in admitting evidence ‘only when no reasonable person could agree’ with its rulings.”) (quoting United States v. Sinclair, 74 F.3d 753, 756 (7th Cir.1996)).
The dissent disagrees with the trial judge’s decision to exclude the affidavit. Initially, the dissent, according to the majority’s interpretation, gives somewhat of an inaccurate accounting of the events that transpired prior to Amerson’s arrest for possession of crack cocaine. The dissenting judge notes that after Amerson was warned that the police were inside the apartment, “Amerson tossed a plastic bag over his shoulder. The police testified that they found the [plastic] bag, containing small ‘rocks’ of crack cocaine, on the ground in front of the building but that they found nothing but shards of glass on the ground below the window that Heard had broken.” On the contrary, the police did not testify that the plastic bag was found beneath the bedroom window as the dissent seems to imply, but rather, the testimony of Officer Moore revealed that Amerson’s plastic bag was discovered by Moore directly below the front porch where Amerson was standing and taken into custody by the officers, some “thirty to forty feet” east and slightly to the south of the broken window.
The dissent also comments: “[ajlthough one officer testified that the crack allegedly thrown by Amerson was found at the front of the apartment, another officer, the [684]*684evidence custodian [Officer Patterson],9 placed it only about 19 feet in a straight line from the window in the back bedroom.” Initially, we disagree with the dissent, for we do not believe it is possible for a person to throw a plastic bag nineteen feet “in a straight line” through a second floor window covered by a heavy curtain. Furthermore, the dissent fails to point out that the testimony the majority relied upon dealing with the location of the crack was that of Officer Moore, the police officer who discovered the plastic bag containing the crack on the ground below the porch. Isn’t it. interesting to note that exhibit 4E received in evidence is a photograph of the bag of crack as it was located at the time of its discovery in the light of Officer Moore’s flashlight on the ground immediately below the porch? In response to the dissent’s contention that the evidence custodian located the bag nineteen feet from the window, we point out that the evidence custodian was not responsible for locating the cocaine, Officer Moore was, so the evidence custodian’s observations are thus of no import.
We also disagree with the dissent’s contention that the district judge failed to give reasons for his ruling that the affidavit was not trustworthy and that his ruling therefore deserves no deference. Specifically, the dissent provides:
On the present, incomplete record, Heard’s affidavit seems about as trustworthy as other forms of hearsay.... The district judge thought otherwise, but as he did not explain his thinking process, and specifically did not point to anything in the record that might support his conclusion, we should not give it controlling weight.... [Djeference is never properly given to rulings the grounds of which are not explained, unless the grounds are obvious.
The judge specifically stated his reason for finding that the affidavit was not trustworthy — he found that the affidavit was not trustworthy because it lacked sufficient corroborating circumstances. The judge went on to state: “The third element [of Fed.R.Evid. 804(b)(3) ] is that there must have been corroborating circumstances which clearly indicate the trustworthiness of the statement. I can not in good conscience find that third element has been met.” (emphasis added). Since the judge, with this very language, did in fact make a specific finding dealing with the lack of trustworthiness of the affidavit, we are obligated to extend considerable deference to his ruling and overturn it only if the ruling was “clearly erroneous.” Hall, 165 F.3d at 1112 (quoting Garcia, 897 F.2d at 1420) (emphasis added). The trial judge may exclude statements where “ ‘the corroborating evidence does not clearly indicate the trustworthiness’ of the statements.... A district court’s ‘determination of the trustworthiness of an out-of-court statement should be upheld unless the finding is clearly erroneous.’ ” Id. Furthermore, Heard’s affidavit is of little value, for Heard never affirmatively stated that he disposed of the crack, only that he “attempted to” do so. Additionally, based on the testimony of Officer Mumaw, Heard was unable to throw anything out the window since “there was a large curtain in the way.” It is interesting to note that the bag of crack was discovered some thirty to forty feet east of the window in question by police after the officers had witnessed Amerson throw it over his head. We query the dissent: how could Heard throw a plastic bag of crack co[685]*685caine out of a second floor window some thirty to forty feet east parallel to the apartment through a heavy curtain unless he was somehow hanging out the window?
The dissent goes on to comment that “no reason has been given why Heard ... would admit to that possession if it were untrue.” We believe that this statement is inaccurate because the dissent apparently overlooks the information contained in Amerson’s presentence investigation report, which provides that Amerson had a criminal record. The jury had been apprised by defense counsel that Amerson had previously been convicted of committing three felonies. As such, Amerson’s credibility might very well have been questioned by the jury, and the court provided jurors with an instruction concerning the jurors’ consideration of Amerson’s prior criminal convictions — “Evidence that a witness has been convicted of a crime is to be considered by you only insofar as it may affect the witness’ credibility.” Armed with this instruction, jurors were prepared to properly assess Amerson’s credibility. Furthermore, Amerson had in the past been accused of using physical force to coerce an associate to sign an affidavit “stating that [the associate] was the owner of ... drugs found in [Amerson’s] vehicle,” thus, a false affidavit which exculpated Amerson and inculpated his associate. Considering the surrounding circumstances that clearly demonstrate that the affidavit was untruthful, it would not be improper for us to believe that Amer-son coerced Heard to sign the affidavit.
The dissent comments: “It is apparent from the majority opinion that my colleagues credit the police testimony over that of the criminals; but that is not our job; the issue is the admissibility, not the weight, of Heard’s affidavit.” In so arguing, the dissent confuses our determination of the issue. By rejecting Amerson’s testimony, we do not give greater weight to the testimony of a single police officer or even two police officers; the fact is that all six officers who testified repeated the same set of circumstances. Does the dissent really believe that all six officers were untruthful? The presiding judge made a credibility judgment based on the testimony of the defendant and his witnesses following the testimony of the law enforcement officers. Further, let us not ignore the fact that Amerson has repeatedly been involved in criminal activity and has previously been accused of coercing an associate to sign a false affidavit. As such, we cannot credit Amerson’s testimony over that of the police. Furthermore, we note that one who has previously been convicted of a criminal offense is not as trustworthy as one who has never been convicted of a crime.
Also, the dissent is inaccurate in contending that the quantity of cocaine which Heard claimed belonged to him, which we concluded that Amerson threw away, would “ice” the case against Heard. Since Heard had already been caught red-handed with a canister of cocaine in the apartment, the Government’s case against him was already “on ice.”
The dissent goes on to state: “[b]e-fore ruling Heard’s affidavit inadmissible, the district judge should have explored the circumstances surrounding its creation.” To the contrary, the burden to demonstrate the existence of corroborating circumstances is on Amerson. See Am. Automotive Accessories v. Fishman, 175 F.3d 534, 541-42 (7th Cir.1999). Shifting the responsibility of demonstrating corroborating circumstances to the district judge is improper since the judge should not become an advocate of one party or another.
B. Prosecutorial misconduct
Amerson next contends that he was denied “due process of law and a fair trial” as a result of the prosecution’s allegedly inflammatory statements at closing argument. “As a general matter, improper comments during closing arguments rarely rise to the level of reversible error, and considerable discretion is entrusted to the [686]*686district court to supervise the arguments of counsel.” United States v. Wilson, 985 F.2d 348, 353 (7th Cir.1993) (internal quotation omitted). A new trial is required only if the improper comments prejudiced the defendant’s right to a fair trial. See United States v. DePriest, 6 F.3d 1201, 1209-10 (7th Cir.1993); see also Darden v. Wainwright, 477 U.S. 168, 181, 106 S.Ct. 2464, 91 L.Ed.2d 144 (1986) (“[I]t is not enough that the prosecutors’ remarks were undesirable or even universally condemned. The relevant question is whether the prosecutors’ comments so infected the trial iuith unfairness as to make the resulting conviction a denial of due process.”) (citations and internal quotations omitted) (emphasis added).
In determining whether statements by a prosecutor rise to the level of a constitutional infraction, we apply the standard endorsed by us in United States ex rel. Clark v. Fike, 538 F.2d 750, 760 (7th Cir.1976) (quoting United States ex rel. Kirk v. Petrelli, 331 F.Supp. 792, 795-96 (N.D.Ill.1971)): “The question to be decided is whether the ... statements were so inflammatory and prejudicial to the defendant petitioner as to deprive him of a fair trial and thus deprive him of his liberty without due process of law as proscribed by the Fourteenth Amendment.”
United States ex rel. Garcia v. Lane, 698 F.2d 900, 902 (7th Cir.1983). In reviewing whether a prosecutor’s statements deprived a defendant of a fair trial, we examine five factors: 1.) the nature and seriousness of the misconduct; 2.) the extent to which the comments were invited by the defense; 3.) the extent to which any prejudice was ameliorated by the court’s instruction to the jury; 4.) the defense’s opportunity to counter any prejudice; and 5.) the weight of the evidence supporting the conviction. See United States v. Kelly, 991 F.2d 1308, 1315 (7th Cir.1993) (citation omitted).
Initially, Amerson contends that the prosecutor improperly vouched for the credibility of the police officer witnesses when he stated during closing argument “[w]hat you’ve got here are vice and narcotics officers — experienced drug enforcement officers whose job it is to arrest dope peddlers ... ”, and “[t]hose officers all indicated that it was part of their duties to rid our streets of cocaine.... ” The trial judge sustained Amerson’s objection, but the defendant contends on appeal that the vouching denied him his right to a fair trial.10
In this Circuit, a prosecutor vouches for a witness by “expressing] her personal belief in the truthfulness of a witness” or by “implying] that facts not before the jury lend a witness credibility.” United States v. Renteria, 106 F.3d 765, 767 (7th Cir.1997) (citing United States v. Robinson, 8 F.3d 398, 415 (7th Cir.1993)). We disagree with the trial judge’s decision to sustain Amerson’s objection to the prosecutor’s alleged vouching (and Amer-son’s contention that the statements of the prosecutor referred to in this case rose to the level of vouching), for the prosecutor did nothing more than explain the duties of the police officers; he stopped short of endorsing a personal opinion about the officers’ testimony. The disputed statements were within the bounds of acceptable argument and under no circumstances should they be classified as amounting to improper vouching for the credibility of the witnesses.
Amerson next alleges that the prosecutor shifted the burden of proof to the defendant when he stated “[w]ho is telling the truth and who is lying here.... You simply cannot believe the testimony of these police officers and believe the defendant’s testimony at the same time.” The trial judge overruled the defense counsel’s objection that the prosecution distorted the burden of proof, ruling that “there [687]*687[was] direct testimony from each side that would make [the comments of the prosecutor] an appropriate comment.” Amerson argues that, under the holding enunciated in United States v. Vargas, 583 F.2d 380 (7th Cir.1978), the prosecutor’s statement was improper. In Vargas, the prosecutor told the jury that
these Federal agents have come in and testified under oath as to what they observed; and if you find the defendant not guilty, I want you to write on [the verdict card] that all of those people lied. [Agent] Ricevuto is a liar. [Agent] Garcia is a liar. [Agent] Collins is a liar. [Agent] Kowalski is a liar. [Agent] Fanter is a liar.
Id. at 387. The prosecutor in that case attempted to instruct the jury that the only way they could find the defendant not guilty would be to find that all of the federal agents were lying. See id. The Vargas court found the comments of the prosecutor were beyond the bounds of proper argument because the prosecutor distorted the burden of proof. See id. Not content to let the jury decide the case according to the judge’s instructions, he set up a “false dilemma” by informing the jury that they had to choose between two and only two options — either the defendant was lying or all the federal agents were lying — when in fact the jury had more options than only those two:
Even assuming that the testimony of the prosecution and defense witnesses contained unavoidable contradictions, it of course does not follow as a matter of law that in order to acquit Vargas the jury had to believe that the agents had lied. If the jurors believed that the agents probably were telling the truth and that Vargas probably was lying or even if the jury was convinced that all of the agents save Garcia were telling the truth and thought that Garcia probably was telling the truth it would have been proper to return a verdict of not guilty because the evidence might not be sufficient to convict defendant beyond a reasonable doubt. To tell the jurors that they had to choose between the two stories was error.
Id. at 387 (citation omitted) (emphases added). The case at hand differs from Vargas because unlike the Assistant United States Attorney (“AUSA”) in Vargas, the prosecutor here did not advise the jury that they were required to choose between two “stark, bright-line, and absolute alternatives,” United States v. Marshall, 75 F.3d 1097, 1108 (7th Cir.1996). Since the AUSA made a mere statement of fact, which was no different than stating to the jury that they had a chance to determine whether the officers or the defendant was telling the truth and that it is up to the jury to determine who was more credible when applying the court’s jury instructions to the evidence received, his comments were not objectionable.
C. Franks hearing
Amerson next argues that the judge erred when he denied the defendant’s pretrial motion for a Franks hearing, alleging that the police officers’ search warrant affidavit contained a false statement. See Franks v. Delaware, 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978). The affidavit stated, in relevant part,
On February 11, 1997, complainant received information from a confidential informant who has given reliable information to complainant in the past stating that informant had been present in the above described premise [sic] on more than three occasions in the past sixty days, the most recent having been within the past 72 hours. On each occasion informant observed a quantity of off-white rocklike substance which appeared to informant to be rock cocaine. Informant has purchased rock cocaine in the past and stated that the off-white rocklike substance informant observed in the above described premise [sic] appeared to be similar to other rock cocaine informant has purchased in the past. On at least one of these occasions the off-white rocklike substance was [688]*688represented to be cocaine by a black male, known to informant as Robert Amerson.... Informant told complainant that Robert Amerson routinely sells crack cocaine from this apartment on a daily basis.
The affidavit alleges that Amerson was present in Tolliver’s apartment sometime in the 72 hours prior to the issuance of the warrant on February 11, 1997, and Amer-son denied that he was in the apartment during that time frame.
The search warrant, issued at 5:10 p.m. on February 11, 1997, reflects that the confidential informant saw Amerson in the apartment sometime between 5:10 p.m. on February 8, 1997, and 5:10 p.m. on February 11, 1997. In support of his claims, Amerson filed three supporting affidavits which he states establish that he was not present in the apartment during that time frame: 1.) an affidavit from his wife stating that the defendant was with her when she rented an automobile at the Peoria airport at 5:35 p.m. on February 8; 2.) an affidavit from friend Mike Diggins stating that on that same day (February 8) Amer-son picked him up at 6:30 p.m. to go to Chicago, and that they did not return until late in the afternoon on February 10; and 3.) an affidavit from Jesse Tolliver, the lessee of the apartment stating that he was out of his apartment from 10:00 p.m. February 9 until 4:00 p.m. February 11, and that no one else had access to the apartment.
The trial judge denied Amerson’s motion for a Franks hearing, finding that even if the proffered affidavits were true, Amer-son failed to proffer an alibi for the entire relevant time frame; in particular, the affidavits did not account for his whereabouts between 5:10 and 5:35 p.m. on February 8. The judge ruled that Amerson failed to make a prima facie demonstration that the confidential informant’s information contained in the affidavit was inaccurate since he fell short of making a “substantial preliminary showing” that the search warrant affidavit contained a falsehood. Franks, 438 U.S. at 155, 98 S.Ct. 2674.
There are two requirements for a Franks hearing: 1.) the movant must demonstrate that the affidavit contained erroneous information; and 2.) the movant must demonstrate that the police officers knew that the affidavit was false or at least demonstrate that the officers recklessly disregarded the truth. See United States v. Pritchard, 745 F.2d 1112, 1119 (7th Cir.1984). “There is, of course, a presumption of validity with respect to the affidavit supporting the search warrant.” Franks, 438 U.S. at 171, 98 S.Ct. 2674. However, if a defendant can make a “substantial preliminary showing” that an affiant knowingly submitted inaccurate material statements or recklessly disregarded the truth, the Fourth Amendment requires that the district court hold an evidentiary hearing. See United States v. Causey, 9 F.3d 1341, 1343 (7th Cir.1993) (citing Franks, 438 U.S. at 155-56, 98 S.Ct. 2674). We review a district court’s denial of a request for a Franks hearing for clear error. See United States v. Hunter, 86 F.3d 679, 682 (7th Cir.1996).
Amerson’s wife’s affidavit established that he and his wife were at the Peoria, Illinois, airport renting an automobile at 5:35 p.m. on February 8 (a time we can ascertain with specificity because the time 5:35 p.m. is inscribed on an Avis rental car receipt issued to his wife). However, the judge noted, “an alibi means that it completely covers the period of time of it,” and Amerson’s alibi does not cover the entire relevant time period since, as noted above, it fails to account for his whereabouts between 5:10 p.m. and 5:35 p.m. on February 8, 1997. Amerson therefore fails to establish an alibi for the complete 72-hour period of time and thus obviously fails to establish that the trial judge clearly erred in ruling that a lapse existed. See id.
Furthermore, even had Amerson succeeded in demonstrating that the information in the search warrant affidavit was false, he failed to comply with the other [689]*689all-important Franks requirement: that the law enforcement officers seeking the warrant had knowledge that the affidavit they filed was false or that they acted in reckless disregard of the truth, see United States v. Pritchard, 745 F.2d 1112, 1119 (7th Cir.1984) (“A Franks violation occurs only if the affiant knew the third party was lying, or if the affiant proceeded in reckless disregard of the truth.”), since he failed to address this requirement in his brief.
D. Sufficiency of the Evidence
Lastly, Amerson requests a new trial on the grounds that “the evidence adduced at trial was legally insufficient to prove him guilty of the alleged offense. Specifically, ... it was physically impossible for him to have projected any bag ... containing 6.4 grams of crack cocaine over his head while standing at the door to the upstairs apartment with the screen door resting at his side and back.” In reviewing a claim of insufficiency of the evidence, this Court “eonsider[s] all the evidence and all reasonable inferences that can be drawn from the evidence in the fight most favorable to the government.” United States v. Hernandez, 13 F.3d 248, 251 (7th Cir.1994). We uphold a conviction if “ ‘any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.’ ” United States v. Saunders, 166 F.3d 907, 912 (7th Cir.1999) (quoting Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979)). Reversal is warranted “ ‘only when the record is devoid of any evidence, regardless of how it is weighed, from which a jury could find guilt beyond a reasonable doubt.’ ” United States v. Garcia, 35 F.3d 1125, 1128 (7th Cir.1994) (quoting United States v. Gutierrez, 978 F.2d 1463, 1468-69 (7th Cir.1992)).
In support of his contention that it was impossible for him to dispose of the bag of cocaine, Amerson merely presents to the Court the statement reciting that “when this Court reviews the testimony and evidence adduced at trial, and examines the photographic exhibits ..., this Court can reasonably conclude that it was physically impossible for Mr. Amerson to have committed the alleged offense in the manner in which the officers contended, and that no rational trier of fact could have reached a contrary conclusion.”11 Amerson’s brief fails to provide any facts, any exhibits, evidence, and/or logical reasoning, thus flouting the requirement in the federal rules of appellate procedure that an appellant must provide the court with “reasoned discussion” supporting an appellate claim. See Doherty v. City of Chicago, 75 F.3d 318, 324 (7th Cir.1996) (“[I]t has long been the law of this circuit that Rule 28(a)(4) [now (a)(6) ] of the Federal Rules of Appellate Procedure requires that an appellant present in its brief the issues it desires to litigate supported by appropriate judicial authority and reasoned discussion.”) (emphasis added) (citing Littlefield v. McGuffey, 954 F.2d 1337, 1342 (7th Cir.1992); F.T.C. v. World Travel Vacation Brokers, Inc., 861 F.2d 1020, 1025-26 (7th Cir.1988) (collecting cases); and Beard v. Whitley County REMC, 840 F.2d 405, 408 (7th Cir.1988)). Given our adversarial system of litigation, “[i]t is not the role of this court to research and construct the legal arguments open to parties, especially when they are represented by counsel.” Doherty, 75 F.3d at 324 (citing Sanchez v. Miller, 792 F.2d 694, 703 (7th Cir.1986)). Because Amerson miserably failed to establish the basis for his allegation that it was physically impossible for him to have thrown the bag of crack cocaine in the manner witnessed and testified to by the officers, we are not obligated to review his claim.
[690]*690IV. CONCLUSION
We are convinced that the trial court did not commit error in refusing to allow the introduction of the affidavit of Timothy Heard since the affidavit was both uncorroborated and untruthful. Also, we are persuaded that the court did not commit error in refusing to rule that the prosecutor improperly distorted the burden of proof during closing argument or in denying Amerson’s request for a Franks hearing. The evidence in the record of Amer-son’s guilt is overwhelming and more than sufficient to convict him of each and every element of the crime charged beyond a reasonable doubt.
AFFIRMED.