In the Gospel of Jesus Christ, the Lord says many times: “He who has ears to hear, let Him hear!”

In the Church today the Pharisees say:

“Though the New Testament is written in Greek, Jesus spoke Aramaic.”

Hold Your Tongue, Preacher!  You’re deceived. Where else have we heard of something very similar to this?  Is it not that the Old Testament was written in Hebrew, even though the script is only ever written in Aramaic?  This is the exact same argument in antithesis.  How scandalous can one become, that a preacher would say to the Son of God: “Even though the WORD of GOD came in Greek, you still do not know the Language of God, nor do you know what it sounds like; because the Language of God and the Sound of God is only ever heard in Aramaic.”  This is the teaching of an unclean spirit; it is foul and it is demonic.

This insidious teaching regarding the Aramaic and Scripture is called “The Zeta Effect.”  It is extremely deceptive and destructive teaching that originates from the Scribes and is preached by the Pharisees.  At its core, this teaching maintains that the sixth (6th) letter of the Greek alphabet is “Z” but that this letter as a numeral is worth seven (7).  It is a very specific type of contradiction aimed at God and God’s Chosen.  The Zeta Effect teaches that the Path of God is crooked.  It claims a type of “rest,”  but it does not give rest; it only gives more contradiction, inconsistency, confusion, lies, and much more.  It is sinister and devilish. It is a game, and a game, moreover, that is played again and again by contemporary Christianity with the World.

To all such games I say “No.”  The sixth letter of the Language of God is  ” C ” (Stau), and as a numeral it is worth six (6).  STAU also doubles as SIGMA  “C” which looks exactly the same as the Stau.

Do not be conformed to this world!  Make straight the Way of the Lord!

I think I’m about ready to tie together Farey sequences and the Riemann zeta function (specifically, the Riemann hypothesis). I know there were lots of gaps along the way, and lots more to explore from here, so that all gets to be the dénouement, if/when I get to it.

Yesterday we ended with the identity

Define , which you might call the cumulative Möbius function. If you are comfortable with your measure theory, you can decide that there is a measure, which I’ll denote , such that . And if you’re ok with that statement, you probably also don’t mind writing the identity for the reciprocal of zeta, above, as

If you then integrate by parts you find that

Now if (we could say something more like ) for some positive real value , then this most recent integral will converge for . Which means that would have no zeroes in the half-plane . Since there are zeroes on the critical line, where , we know that no value of less than 1/2 will work. But if you can show that for every positive , then you’ll have proven the Riemann hypothesis (and earned yourself a cool million).

Ok, so now we just have to relate the Farey sequences to this somehow.

It’s been a while, so perhaps we should recall some basics about the Farey sequences. The -th Farey sequence, denoted consists of the reduced fractions in with , ordered as usual. For today, let’s actually remove 0 from the sequences. Thinking about how many terms are in , we quickly determine that there are , where is the number of positive integers less than that are relatively prime to (with ). Let me call this sum , so that has terms.

To keep my notation under control, fix an . Then for , let be the -th term in the Farey sequence . The terms in are not equally spaced in , so let us also consider the sequence of equidistant points in . These will be the values . We’re interested in how far the Farey sequence is from this equidistant sequence, so let’s set .

If , then you can show (essentially by Möbius inversion) that

The idea is that the function that is 1 for and 0 for is also

because you can write (fairly cleverly, I feel)

By the way, I’m being a bit loose with my values of these stepwise things when they actually make their steps. Apparently the convention is to define the value at the change to be the midpoint of the values on either side. I feel like it’s a technicality I’m not hugely interested in right now.

Now we apply this identity to , obtaining

That inner sum on the right-hand side is the sum of unit vectors equally distributed around the unit circle, and so is 0 except when . So we obtain

Finally, replace with . After shuffling some symbols around, you can write

Taking absolute values on each side, using the triangle inequality, and the identity we already used about sums of equally spaced unit vectors, we can write

Using our lemma from earlier, about how to obtain the Riemann hypothesis from , we have, at last:

If is for every , then the Riemann hypothesis is true.

Hurray! The result that encouraged me to wander down this path!

So, that was fun. It turns out that both of today’s results that imply the Riemann hypothesis are actually equivalent to the Riemann hypothesis, but perhaps that’s a topic for another day…

[Update 20091203: Noticed I had some absolute value bars in the wrong places in the line before the final theorem, and corrected them (hopefully)]

Yesterday I mentioned Möbius inversion, and I decided that it was worth looking at a little bit more. First of all, it’s pretty fun. Secondly, it’ll give us a useful identity concerning .

Before I get too far, I should add another reference to my growing list. With Edwards’ book, and Hardy and Wright, include Apostol’s “Introduction to Analytic Number Theory.”

Anyway, yesterday I defined the Möbius function

and stated Möbius inversion as follows:

Thm: If then .

Let’s begin today with a proof of this statement. Pf: Suppose is as stated, and consider . By assumption on , this is which I’ll write as . Now in this sum, what is the coefficient of ? Well, it is . We saw yesterday that this sum is 0 unless , i.e., unless . So , as claimed.

That’s pretty fun, I think.

There’s another way to go about these things, and it is by way of what are called Dirichlet series (according to Wikipedia, Dirichlet had many names). Suppose is a function defined on the positive integers (such a function seems to sometimes get called an arithmetic function). I’ll say that the Dirichlet series for is the series . I’ve written it this way, so that I have a function defined whenever the series converges, but I’ll pretty much ignore convergence aspects in what follows.

Suppose and are the Dirichlet series associated to some , respectively. What does the product of these two series look like? Well, since , you can determine that the product of these two Dirichlet series is another Dirichlet series, this one associated to the function . Indeed, just ask yourself, “what is the coefficient of in the product of the series and ?”

By the way, I’ve called the product series because it also gets called the convolution of and . The convolution of and is denoted , and is simply the function .

Now you’ve got an operation, , on arithmetic functions, and you can start asking what properties it has. It turns out that gives the set of arithmetic functions (well, those functions that aren’t 0 on the input 1) the structure of a commutative group. The identity is the function with and if (corresponding to the identity Dirichlet series, the constant function 1). It’s sort of a fun little induction problem to find a formula for the inverse of a given arithmetic series. We won’t need the formula exactly, so perhaps I’ll leave it for you.

Let , so that the associated Dirichlet series is .

We now have another language in which to state Möbius inversion:

Thm: If then .

In this statement, let (the identity of convolution) so that . Then we obtain . On taking Dirichlet series, we find that

or, in other words,

How’s that for an identity?

There’s another way to get at this identity, without the language of Dirichlet series (explicitly anyway), which I wrote “holy crap” next to in the book (Hardy and Wright) when I read it. So probably I should share. Using Euler’s product (recall the indexing is over primes)

we have

Here’s the part that rocked me: Look at and write it as , i.e. as . Because the first thing I think to do when I see a binomial is to write it as an infinite series of mostly 0s. Like I said yesterday, I guess it helps if you know where you are going. So anyway, we’ve got

Now, like we’ve been doing, let’s ask, “what is the coefficient of when the product is all written out?” Well, the only way to get is the term corresponding to the prime factorization of . If , then the coefficient is . But is “multiplicative,” meaning that if and are relatively prime then . So the coefficient of is just . This proves the identity.

That’s the identity we’ll use in the next post or two to finally tie together Farey sequences and the zeta function, believe it or not. So I’ll basically stop here. But I do want to mention that you can do some of the things we’ve done above in slightly more general contexts, and I remember really enjoying reading about them. Namely, you can work in other posets. In what we’ve been doing above, there has, somewhere in the background, been the poset of positive integers ordered by iff . Probably a good place to start reading about the generalization is Wikipedia, as usual. If I find, sometime, the paper I read where I first saw these things, I’ll update this post.

In the mean time…

Yesterday, I related the logarithm of to a piecewise linear function . You may recall that was defined for positive reals by setting it equal to 0 at , and then jumping by whenever , for some prime and integer . At the end of the day, we got to

where . Today, we’ll analyze some more, and re-write the formula above.

When I introduced , I ended with the following formula:

where the bounds on that integral are supposed to represent a curve that “starts” at the right-hand “end” of the real line, loops around 0, and then goes back out the positive axis to infinity. I’m not good enough at complex line integrals at this point to say any more about this. But apparently if you are good at these sorts of integrals, using Cauchy’s integral formula and things you can find the so-called “functional equation”

If you then use the relation I mentioned previously:

(well, you use this for ), and one I haven’t mentioned:

and move some symbols around, you arrive at a more symmetric equation:

Notice that if you plug in the formula on the left-hand side, you obtain the right-hand side.

This function on the left-hand side apparently has poles at 0 and 1, so if we define

then we obtain an entire analytic function satisfying . Using the factorial relation for , we can re-write as

I get the impression that if you know what you are doing, then the things above aren’t tooo hard to justify. Apparently the next part is a bit trickier. Apparently, you can write

where the product is indexed over the roots, , of (so ).

If you’ve heard anything about the Riemann hypothesis, you know that the roots (the “non-trivial” ones, I didn’t talk about the trivial ones) of are a big deal. Our second formula for shows that they are (basically) the same as the roots of , and so they are the that the sum above is indexed over. The symmetric equation from earlier has a little something to say about the zeroes, and it has been shown that all of the zeroes have real part bigger than 0 and less than 1 (this is called the “critical strip”). The hypothesis (whose truth won’t affect what we’re saying below) is that all of the zeroes have real part 1/2 (this is the “critical line”). Apparently Riemann didn’t need this hypothesis for the things in his paper that introduced it, so I don’t really have much more to say about it right now. Although, honestly, I still don’t see what all the fuss is about The formulas we’ll get below and tomorrow work even if the roots aren’t on the critical line (unless I’m missing something important. If I am, please comment).

Anyway, back to the topic at hand. Let me try to convince you that it isn’t horribly unreasonable to think about writing a function as a product over its roots, as I’ve done above. For the sake of example, let (or pick your own favorite polynomial). The usual way this would get factored, in all the classes I’ve ever taken or taught, is (up to permutation) , showing that the roots are . However, if you factor a 4 out of the term, and -1 and -2 out of the other terms, you can also write . You still see all the zeroes when you write the polynomial this way. You can also see that the coefficient in the front is . So we’ve written , which is the same goal as what we’re doing with above. Incidentally, the idea of writing a function this way was also used by Euler to establish (I’ve mentioned this briefly elsewhere).

We now have two formulas for , so we can put them together to get

Recalling that our formula for , at the beginning, involved , let’s take the log of the equation above and solve for the term:

The idea is now to plug this in the formula for . Apparently if you do, though, you’ll have some issues with convergence. So actually try to do the integral in , using integration by parts (hint: ). The “” term goes to 0 and you obtain

where, as before . Now plug in the 5 terms we’ve got above for , and you get a formula for . What happens to the terms? Can you actually work out any of the integrals?

Well, you might be able to. I’m not. Not right now anyway. But I can tell you about what others have figured out (rather like I’ve been doing all along, in fact)…

It’s clear that the term drops out, because you divide by and then take the derivative of a constant and just get 0. The term with ends up just giving you , which is .

The term corresponding to the term with in it can be rewritten as

(as if that were helpful).

The important terms seem to involve the function

Of course, this integrand has a bit of an asymptote at 1, so really (in Edwards’ book, anyway) is the “Cauchy principal value” of this integral, namely

This function is, rather famously, related to approximating the number of primes less than a given bound. In fact, tomorrow I plan on having more to say about this. But back to the terms in our integral for .

The term corresponding to the sum over the roots ends up giving you

.

But apparently the dominant term is the term corresponding to . It actually gives you

So, finally, we have written

Doesn’t that make you feel better? We started with the reasonably understandable

,

and created the monstrosity above. I guess this is why I’m not an analyst. To me, it seems worse to write as this terrible combination of lots of integrals. But apparently it’s useful in analysis to have such formulas. I guess we’ll see a use tomorrow…

Last time (too long ago, sorry), I finally got around to talking about the function . Today, I think I’d like to talk about .

First, though, I should mention Euler’s product formula. The goal is to re-write the infinite series as a product. For each prime , we can tease out of the series above those terms where is to some power. That is, we can think about the sub-series . This is a geometric series that converges to . If we take two such series and multiply them together, which terms from our starting series for will we recover?

Suppose and are two primes, and consider the product

.

The terms in that we obtain from this product are those where is a product of some power (possibly the 0 power) of and some power (again, possibly 0) of . We could then think about another prime, say , and it’s geometric series, and multiply it by the above, and obtain all the terms in where for some .

Continuing on, and recalling that every positive integer has a unique factorization as a product of primes, we obtain the magical formula (isn’t most of Euler’s work magical?):

,

where the product is indexed by the primes. What’s nice about this, for us, at the moment, is that logarithms work well with products and powers, but not soo well for sums. Recalling the Taylor polynomial

we find

.

That was fun. Incidentally, it’s not much of a jump from here to show that the series of prime reciprocals diverges. I mentioned it in a post a while ago.

Let’s switch gears for a little bit. I’m going to define a function on the positive reals. If you’re following Edwards’ book with me (as I jump all over the place), this will be almost exactly his , differing only at the integers (for today anyway ). Let me start by setting . The function will be a step function, which I’ll define to mean piecewise linear with slope 0 on each piece. So to define I only need to tell you when to jump, and by how much. The rule is: when you get to an integer , if for some prime , then you jump up by . So at you jump by 1, at you jump by 1/2, at you jump by 1/3, etc. Here’s a little picture of , with a few values highlighted:

Slightly more precisely, we can write

.

Now, let me see if I can convince you that there is some justification in writing

.

Let’s work with the right-hand side. Since near , I’ll actually start my integral at . I think the way to go about it is as follows:

Now suppose that for some . Then the terms corresponding to and telescope as follows:

.

If, also, , this we can telescope another term into this one, and so on. So, really, the important in this sum are those where jumps, which are the prime powers. Let be the -th prime power (i.e., the point where makes its -th jump), starting with . Then we can write

What good is writing this way? I guess if you know something about Fourier inversion (which I don’t) then you get to say that

for . What good is that? I think I’ll have to read some more and tell you about it tomorrow, but it’ll turn out to be useful once we have yet another description of , in terms of the 0s of (finally getting to those 0s everybody cares so much about).

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I guess it is about time to get to the zeta function side of this story, if we’re ever going to use Farey sequences to show how you could prove the Riemann hypothesis. I’ve been reading a bit of Edwards’ book, with the same title as this post, and thought I’d try to summarize the first chapter over the next few posts. I’m not sure that the content of the first chapter is hugely vital for the final goal of relating to the Farey sequences, but I wanted to try to learn some of it anyway.

I should mention, before talking about any of this, that I will not claim to know any complex analysis. The last complex analysis course I took was 5 years ago, I can’t be sure how much I paid attention at the time, and I haven’t used it since. I will be jumping over gaps of various sizes for quite a while in the upcoming posts. Perhaps sometimes I’ll mention that a gap is there. Mostly, though, what I’m after in this story is the outline, and how all of the parts fit together, as a big picture. Perhaps I’ll go back and fill in some gaps, as I understand more.

For today, let’s see if I can build up to an analytic expression of . Our starting point is the function

which is defined for real values of larger than 1. The goal is to find a nice expression that is defined on more of the complex plane, but agrees with this definition on the reals larger than 1.

To get to that point, we’ll use the function:

This is an analytic function defined everywhere except at the negative integers and 0. If you are only interested in real values of , this function is a continuous extension of the factorial function, which is only defined on the positive integers. Actually, there is a shift involved, so (Edwards blames this shift on Legendre, whose reasons, he states, “are obscure”. Edwards uses throughout, so I might make some errors). In particular, when both sides make sense.

Another relation that we’ll use is that

If memory serves (from reading Artin’s lovely little book on the function), you can obtain this by determining that the derivative of the expression on the right is 0. This tells you that is a constant, and so you can pick your favorite value of to obtain the value of that constant. Anyway, the identity I’ll use is a rearrangement of the one above, namely

.

Now, in the expression

,

substitute for some positive . Messing about with that substitution for a few minutes (and then re-writing by abuse of notation), you can arrive at

.

That in the denominator is useful for us, as that’s how it shows up in . In particular, we can obtain

That last transition coming about by summing the geometric series. There are probably some things analysts like to check in here, moving infinite sums and improper integrals past each other… I’ll let them.

Ok, we’re nearly there. Next up, you do some complex line integral and show that

.

That integral is a little weird, going “from to “. Really, we take a path that “starts at “, swings around , then goes back out to infinity. This is almost certainly one of the complex analysis things I should go back and learn more about. Since we are integrating along positive real values of , we’re working with logs along the negative real axis. Perhaps I’ll return to this integral in a future post.

Using the identity about from above, we obtain

.

We’re now at a point where we’ve (apparently) got an expression for that is analytic in the complex plane except for a simple pole at .

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Ziarul CanCan sustine ca Virgil Iantu o inseala pe iubita sa, producatoarea tv Roxana Alexandru, cu actrita Mirela Zeta, cea care o joaca pe Andreea Marin in scenetele trupei Mondenii.

Conform CanCan, Virgil Iantu si Mirela Zeta au apelat la ajutorul lui Vlad Enachescu pentru a-si face rost de o camera in noaptea de 9 spre 10 octombrie, intr-un hotel din nordul capitalei. Vlad Enachescu a achitat cu cardul personal contravaloarea camerei pentru o noapte, 100 de euro, iar apoi i-a dat cheia lui Virgil care astepta, alaturi de Mirela, in afara hotelului. 　Cei doi colegi de la Prima TV au urcat ulterior in camera cu pricina, unde au petrecut 40 de minute. Primul care a coborat a fost Virgil Iantu, carea incercat sa treaca neobservat insa a fost oprit de receptioner pentru un autograf, iar dupa cinci minute a coborat si Mirela Zeta care a preferat sa lase capul in pamant si doar sa predea cardul de la camera

Enfin ! clament tous les mordus de ce jeu de rôle post-apocalyptique. Vendredi 16 octobre sortait l’édition Jeu de l’Année du hit de Bethesda. Une version qui comprend le jeu de base ainsi que ces cinq extensions.

Des mutants, de l'humour noir et du Nuka-Cola.

Les développeurs l’avaient annoncé : Mothership Zeta serait la dernière extension de Fallout 3. Depuis la sortie en octobre 2008 du jeu original, les petits gars de Bethesda n’ont pourtant pas chômé. Le studio a développé quatre autres extensions : Operation : Anchorage, The Pitt, Broken Steel et Point Lookout. Seul bémol, il fallait débourser près d’une dizaine d’euros pour un seul de ces contenus téléchargeables. Depuis vendredi, ils sont donc réunis sous boîte avec le jeu de base, pour une somme de 60 euros sur X360 et PS3 et de 50 euros sur PC. À essayer si vous n’avez pas encore eu l’occasion d’errer dans un Washington D.C. ravagé par un conflit nucléaire.

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By now, you know all there is to know about the not-for-public-consumption, Camaro-engined cop Caprice. Wannadrive wouldn’t normally be concerned with a V-8 American sedan, except this one isn’t. It’s the technological successor to the superb, dead Pontiac G8, and Australian masterpiece, and America doesn’t have another GM V-8 sedan that isn’t a Cadillac.

So Wannadrive went out and scored this exclusive video of it in action–forget worrying about the headlights in your mirror. If you hear this thing roar, just drop to the ground. You haven’t got a chance. Full press release after the jump.

DENVER – An all-new Chevrolet Caprice Police Patrol Vehicle (PPV) will join the ranks of law enforcement departments across North America in 2011. It’s a modern, full-size, rear-drive sedan that will offer both V-8 and V-6 engines, as well as a host of specialized equipment and features.

Chevrolet made the announcement at the annual International Association of Chiefs of Police convention, in Denver, Colorado. The Caprice PPV will be available for ordering next year and will hit the streets in early 2011.

“The new Chevrolet Caprice police car is the right tool at the right time for law enforcement,” said Jim Campbell, general manager for GM Fleet and Commercial Operations. “We asked for a lot of feedback from our police customers, which helped us develop a vehicle that is superior to the Crown Victoria in key areas.”

Vice President, Global Chevrolet Brand Brent Dewar added, “Along with Impala and Tahoe, the Caprice PPV gives agencies a greater range of choices for police and special service vehicles that are all available from Chevrolet.”

Unlike other police cars on the market, the Caprice PPV is not based on existing “civilian” passenger-car model sold in North America. It has been developed in key areas specifically for police duty, containing modern equipment and features:

• Powerful 6.0L V-8 with fuel-saving Active Fuel Management technology and E85 capability delivers expected best-in-class 0-60 acceleration (sub six seconds) and top speed; a V-6 engine will also be offered, beginning in the 2012 model year
• Optional front-seat-only side curtain air bags allows a full-width rear-seat barrier for greater officer safety
• Two trunk-mounted batteries, with one of them dedicated to powering various police equipment
• Designed for five-passenger seating, meaning the upper-center section of the dashboard can be used for equipment mounting without the concern of air bag deployment interference
• Compatibility with in-dash touch-screen computer technology
• Special front seats designed for the long-term comfort of officers whose car is their effective office, including space that accommodates the bulk of a typical equipment belt

The front seats are sculpted to “pocket” the equipment belt, which greatly increases the comfort for a great range of police officer sizes. The foam density of the seatback and cushion insert surfaces are designed to conform to the shape of an equipment belt’s various items, too, allowing the officer’s back to rest properly on the seatback surface.

“The Chevrolet Caprice PPV’s seats represent a revolution in comfort and utility for officers who spend long hours in their car,” said Bob Demick, lead seat design manager. “The shape also enhances entry and egress, making it easier for officers to exit the vehicle quickly. The seatback bolsters, for example, have been purposefully contoured to help pocket the equipment on the belt, which includes the gun, Taser and handcuffs, which rest comfortably in the sculpted lower bolsters. That also increases the longevity of the trim cover surface.”

Along with comfort, the materials used in the seats were also carefully selected. High-wear materials were chosen to stand up to long hours of everyday use, while breathability, long-term durability and ease of cleaning were also important criteria.

Engineers worked on several iterations of the seat, testing a couple of versions in the field to get real-world feedback from police officers, who used prototype seats in their cruisers for a month. Their input helped determine the final design.

Class-leading space

The Caprice PPV is based on GM’s global rear-drive family of vehicles that also underpins the Chevy Camaro. It uses the longest wheelbase of the architecture – 118.5 inches (3,010 mm) – along with a four-wheel independent suspension that delivers responsive high-performance driving characteristics that are crucial in some police scenarios.

Caprice PPV’s long wheelbase also contributes to exceptional spaciousness. Compared to the primary competition, its advantages include:

• A larger interior volume – 112 cubic feet / 3,172 liters – than the Ford Crown Victoria, including nearly 4 inches (101 mm) more rear legroom
• The barrier between the front seat and rear seat is positioned farther rearward, allowing for full front-seat travel and greater recline for officer comfort
• At 18 cubic feet (535 liters) free space (beyond battery located in trunk), the Caprice’s trunk volume is large enough to accommodate a full-size spare tire under a flat load surface in the trunk storage area.

The Caprice’s 6.0-liter V-8 is rated at an estimated 355 horsepower (265 kW) with an estimated 384 lb-ft of torque. It is backed by a six-speed automatic transmission that is performance-calibrated for police duty. Additional, police car-specific powertrain and vehicle system features include:

• High-output alternator
• Engine oil, transmission and power steering coolers
• Standard 18-inch steel wheels with bolt-on center caps
• Large, four-wheel disc brakes with heavy-duty brake pads
• Heavy-duty suspension components
• Police-calibrated stability control system
• Driver information center in the instrument cluster with selectable speed tracking feature.

A host of complementary features are also offered, including special equipment packages such as spotlights; lockouts for the power windows and locks; and an “undercover” street-appearance package (9C3).

To enable more room for interior equipment, the standard radio can be relocated to the trunk, allowing for an in-dash, touch-screen computer to be used.

Caprice on patrol: A brief history

Chevrolet’s history with law enforcement is almost as old as the brand itself. Police departments have used Chevy sedans as police cars for decades, ordering them with basic equipment and powerful V-8 engines – including some special engines that weren’t available in regular-production models, such as the 1959 Biscayne that was offered with up to 315 horsepower.

The full-size Chevrolets joined the force in 1976. All Caprice police cars – including the new, 2011 model – have carried the 9C1 order code. Here’s a quick look back at Chevys on patrol:

1959 – Chevy Biscayne police model capable of 135 mph with specially tuned, police-only version of the 348-cubic-inch V-8 engine

1965 – The new “big-block” 396 engine is offered in Biscayne and Bel Air police cars, making them among the most powerful on patrol; a 427 V-8 was added in 1966

1976 – The 9C1 order code is given for the first time to a full-size Chevy police car package. It carries the Impala name.

1977 – The full-size Chevy is downsized. The 9C1 police package is retained, as is the Impala name.

1986 – The Caprice name replaces Impala, as the car is updated for the mid- and late-1980s – including the option of a powerful, 5.7-liter small-block V-8.

1991 – A new-generation Caprice is launched, with the 9C1 police car still on the beat.

1994 – The 260-horsepower (194 Nm) LT1 V-8 engine is offered in the Caprice 9C1, making it one of the fastest full-size police cars ever offered.

1996 – Caprice police car production ends, as GM’s full-size, body-on-frame car architecture is discontinued.

2011 – The Caprice PPV returns to active duty.