Applicant(S) Information A P P #1 L A P P #2 L A D D R E S S Last Name First Name M: Fill & Download for Free

POS machine is just that. Locked down. If it allows any external input it is completely insecure and responsible for millions of identity thefts.

The original Riemann Hypothesis concerns the location of the (non-trivial) zeros of the Riemann zeta function [math]\zeta(s)[/math]. Among other things, those zeros can be used to describe the distribution of prime numbers, and the Riemann Hypothesis (RH) therefore has consequences for our understanding of this distribution.Deep and profound as it is, the Riemann Hypothesis is about one single function, and correspondingly it is about the distribution of the primes as a whole. But rather than look at the primes as one uniform whole, we can classify them in various ways, and one of the simplest and most natural classifications is to consider the residues they leave modulo various numbers.Here are the primes from 1 to 100:Except for 2, each of them is either 1 less or 1 more than a multiple of 4. Here they are again, colored to indicate which ones are which:The blue ones are 1 more than a multiple of 4, and the orange ones are 1 less. How frequently do they show up? Do they keep showing up? Is it with the same frequency, or is one kind more common than the other?Here is another classification, now looking at multiples of 6. Except for 2 and 3, every prime is either 1 more (green) or 1 less (yellow) than a multiple of 6.Or imagine we classify numbers according to the remainder they leave upon division by 7. No primes except 7 are divisible by 7, but they could potentially leave any one of the other six possible remainders.So now we have six different kinds of primes. Once again, do they all keep popping up? Equally frequently, or not? How accurately can we portray their frequency?Just like the Riemann zeta function can be used to study the distribution of primes, there are functions called Dirichlet [math]L[/math]-series which are used to study these more refined questions.The Riemann zeta function is defined for numbers [math]s[/math] with real part greater than [math]1[/math] like this:[math]\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{for $\Re(s)>1$}[/math]The interesting stuff actually occurs for values of [math]s[/math] that are not covered by this formula, but let’s set this aside for a second. The key to the connection between [math]\zeta(s)[/math] and prime numbers goes back to Euler, and it is this:[math]\displaystyle \zeta(s) = \prod_p (1-p^{-s})^{-1}[/math]The product over all the prime numbers [math]p[/math][math][/math], and each prime contributes a factor of [math](1-p^{-s})^{-1}[/math]. This follows immediately from the Fundamental Theorem of Arithmetic and the sum formula for geometric series.Now, we want a formula which somehow sifts out just those primes that are congruent to [math]1[/math] mod [math]4[/math], or to [math]3[/math] mod [math]7[/math], and so on. We might initially try something like this: define a function [math]f(n)[/math] so that [math]f(n)=1[/math] if [math]n[/math] is in the congruence class we want, and [math]f(n)=0[/math] otherwise. This is the “filter” we want. Now, can we expect that[math]\displaystyle \sum_{n=1}^\infty \frac{f(n)}{n^s} \overset{?}{=} \prod_p (1-f(p)p^{-s})^{-1}[/math]The answer is No, this doesn’t usually work. The problem is that the function [math]f[/math] is often non-multiplicative: for example, if [math]f[/math] is supposed to catch the congruence class [math]3[/math] mod [math]4[/math], then [math]f(7)=f(11)=1[/math] but [math]f(7 \times 11)=f(77)=0[/math]. So [math]f(ab) \ne f(a)f(b)[/math].Dirichlet fixed this by focusing on Dirichlet characters. Those are functions which, like our naive [math]f[/math], are only dependent on the congruence class of their argument modulo some “skip”, but they are multiplicative. For example, consider[math]\chi_0(n) = \begin{cases} 0 & \text{if $n$ is even} [/math][math][/math][math]\\ 1 & \text{if $n$ is odd}\end{cases}[/math][math]\chi_1(n) = \begin{cases} 0 & \text{if $n$ is even} [/math][math][/math][math]\\ 1 & \text{if $n \equiv 1 \pmod{4}$} [/math][math][/math][math]\\ -1 & \text{if $n \equiv 3 \pmod{4}$} \end{cases}[/math]Those functions are characters mod 4, but instead of going 0100 0100 0100 (for the residue 1 mod 4) and 0001 0001 0001 (for the residue 3 mod 4) they go 0101 0101 0101 and 010(-1) 010(-1) 010(-1), and the cool thing is that both of these patterns are multiplicative. In both cases, [math]\chi(mn)=\chi(m)\chi(n)[/math], and that’s true in general for all Dirichlet characters.This is sufficient to restore Euler’s formula:[math]\displaystyle \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p (1-\chi(p)p^{-s})^{-1}[/math]is now true. We, therefore, give those functions a name:[math]\displaystyle L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p (1-\chi(p)p^{-s})^{-1}[/math]We can now try to walk in Euler and Riemann’s footsteps to extract information about primes in those congruence classes using the Dirichlet [math]L[/math]-functions [math]L(s,\chi)[/math].The first victory in this direction was achieved by Dirichlet himself, who managed to prove this beautiful theorem: all the color classes are infinite. Every arithmetic progression whose “shift” is relatively prime to its “skip” contains infinitely many primes.For Riemann, this is merely the assertion that there are infinitely many primes, which of course is elementary; but now when we look at those congruence classes it is (usually) no longer elementary at all. Dirichlet needed to rely on the apparatus of complex function theory and analytic continuation to carefully study his [math]L[/math]-functions and reach the desired conclusion.Furthermore, he was able to provide some basic estimates for the frequency at which primes show up in the different color classes. They basically behave as you’d expect: the same proportion is allocated to each class.And, just like Riemann’s Hypothesis on the location of the zeros of [math]\zeta(s)[/math] gives (among other things) tight bounds on the estimates of the Prime Number Theorem, there is a corresponding hypothesis for each function [math]L(s,\chi)[/math] which, if true, would provide tight bounds on the distribution of primes in congruence classes.This is exactly the Generalized Riemann Hypothesis: the functions [math]L(s,\chi)[/math] (which, like [math]\zeta(s)[/math], can be continued to the whole complex plane by analytic continuation) satisfy [math]L(s,\chi)=0[/math] only when [math]\Re(s)=\frac{1}{2}[/math] or [math]s[/math] is a negative real number.If true, this provides the following tight bound on the number [math]\pi(x,r,k)[/math] of primes less than [math]x[/math] which are congruent to [math]r[/math] mod [math]k[/math]: for any [math]\epsilon>0[/math],[math]\displaystyle \pi(x,r,k) = \frac{1}{\varphi(k)}\int_2^x \frac{dt}{\ln(t)} [/math][math][/math][math]+ O\left(x^{1/2+\epsilon}\right)[/math].([math]\varphi(k)[/math] is the number of residues relatively prime to [math]k[/math], which is simply the number of “color classes” mod [math]k[/math]).The Generalized Riemann Hypothesis (GRH) subsumes the (“ordinary”) Riemann Hypothesis, which is the special case when the character [math]\chi[/math] is trivial, and pushes it much further to control the behavior of primes in those “color classes”. Many applications of RH actually require the stronger GRH; for example, we believe that the Miller-Rabin primality test runs in polynomial time, but we can currently only prove that assuming GRH.There are additional, deeper generalizations of zeta functions to other contexts, in some of which we even know the corresponding RH to be true. But we should leave that for another time.

When I conduct a car stop, I approach and request some form of identification. I prefer a driver's license, but lacking that I can work with a Social Security number or even just a name and a date of birth (bear in mind, though, that if an adult claims to not know their SSN, they're getting some serious scrutiny - this was more often than not the hallmark of someone trying to conceal their identity).With that information in hand, I return to my patrol car. From here, procedures will vary by agency, but in telling you my process know that it will be largely the same anywhere you go. I would first get on an administrative radio channel set up for this purpose and "run your information" through the dispatcher. This consisted of giving your name, date of birth, gender, race, and driver's license or SS number over the radio. The dispatcher queries their systems with this information, and checks for multiple information items:Driving Status. The Department of Revenue database is searched for your information. This record will include all the information present on your driver's license, a current driving privilege status (valid, suspended, revoked, or no privilege), and a list of traffic convictions on your record.Probation or Parole Status. If a driver is on either probation or parole, all law enforcement contacts must be documented and relayed to their P&P officer. If this was the case, dispatch would advise me of this status, give the applicable offense (very important - there will be a big difference in my proceeding actions depending on whether you're on probation for speeding or on parole for assault on a law enforcement officer), and ask for a 'response' - a message to relay to the P&P officer about your demeanor during the contact. I'll then give a very brief description of the contact - usually along the lines of, "Stopped for expired plate, cooperative, issued warning/citation and released," or whatever applies.Caution Indicators. If you have documented instances of being violent during police contact, carrying a weapon, or resisting or assaulting officers, dispatch will notify me using a specific code. This gives me forewarning about previous police contact and may prompt me to request backup or exercise even more acute caution.Warrant Status. Dispatch will query local, state and federal databases with your information to see if there are any outstanding warrants for your arrest:If so, they will see if there is an extradition limit associated with the warrant. This is the maximum distance the issuing agency will travel in order to come and get you from my jail - and depending on the charge it could be as little as a few miles or as much as nationwide.If a warrant comes up and I am within the extradition radius, dispatch will ask if I want the warrant "confirmed." I cannot arrest you based on a warrant hit in a computer system - in order for it to be actionable, dispatch must call the issuing agency on the phone and relay the appropriate information. That agency must then physically pull the paper warrant at their facility, ensure that it is still valid, and confirm this with my dispatcher. Arresting you deprives you of your liberty, which is guaranteed at the most basic level of our governance, so they have to make absolutely positive that this information is still valid; a judge could have rescinded the warrant recently before it could be purged from the system, or any number of other scenarios rendering the hit invalid.So why do they ask if I want it confirmed? The confirmation process will entail the issuing agency clearing the warrant with a note that an arrest was made; dispatch needs to confirm that the suspect is present and that I'm not just running the information in their absence. Also, there are very rarely times when I do not want to confirm a warrant based on circumstances at the scene. If a warrant is confirmed, I am obligated by law to arrest the person - if not, I have just committed a misdemeanor myself. There were very rare instances in which I told dispatch to disregard due to extenuating circumstances (I once declined confirmation on the mother of a week old baby when I found, at two in the morning, that she had a warrant for a parking violation with a $25 bond). You cannot do this lightly and must provide a reason for why the warrant was not confirmed - because a warrant is, by definition, a court order issued by a judge which says that as a law enforcement officer I "shall" arrest the person in question upon lawful contact.While dispatch is working on all of this, I will input your information into the laptop in my patrol car. I will then be able to see most everything dispatch does (in my case, just about everything except local warrants). In the case of your not having hard ID, I can search by your SSN, and if you've got a license I'll be able to see the information on it and get your DoR picture to confirm your identity. I'll also run your license plate through my computer, which will tell me whether the plates or the vehicle are listed as stolen and what vehicle they're supposed to be on.Once dispatch returns their information, I'll wrap up the stop accordingly. If there's a warrant, I will of course get another unit and begin the arrest process. If not, I'll decide whether to issue any citations, then return to your vehicle to return your paperwork and release you with either a warning or your citation(s). Law enforcement personnel enjoy what is termed "officer discretion" during a vehicle stop - they have personal latitude in whether or not to enforce a given law/ordinance in a given situation. In your listed circumstance, the officer decided not to ticket you for the stop sign or the seat belt violations, so once he ensured you were not wanted, he released you. This was probably the case in 40-60% of my traffic stops. I often initiated traffic stops knowing that, barring unforeseen circumstances, I was only going to issue a warning. This served several purposes:Reminding you to address the issue. Just because I don't issue a citation doesn't mean I have no interest in the situation at all. If your plate is expired by a month, I may not issue a citation, but the stop serves as a reminder that you'd better get it renewed sooner rather than later.Free warrant and DWI and [insert offense here] check. I have discovered many, many things inadvertently in the course of a minor probable cause stop - stolen property from a defective headlight, DWIs from failure to use a turn signal, a rape warrant from an expired license plate. Sometimes you have to fish a little - it may take fifty relatively minor stops to get something like that, but all of the intervening warnings were worth the trouble in those cases.Increased law enforcement presence. There's a reason we don't turn our lights off after we've gotten you stopped, even when there's no traffic danger. You know what happens when you see red and blue lights as a driver - you instinctively slow down and drive cautiously, if only for a mile or two. Multiply that by the number of cars passing my stop, and I've affected quite a swath of traffic. Also, criminals tend to keep their heads down with increased police presence. When I'm lighting the trees in a neighborhood up with my strobes, evildoers don't know it's just for a burnt out license plate lamp - they just know the cops are around. That might affect their itinerary for the evening.These were all factors at play in your stop. Thus, in getting a warning for your offenses, you really were a VIP - but only toward the end of increasing our perceived presence in the area you were stopped in. Your assistance is appreciated. :)