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Oct1106, 03:01 PM

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May 23, 2005 This Week's Finds in Mathematical Physics  Week 216 John Baez There are lots of different things called "zeta functions" in mathematics and physics. The granddaddy of them all is the Riemann zeta function: zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + .... This is deeply related to prime numbers, thanks to Euler's product formula zeta(s) = product 1/(1  p^{s}) where we take a product over all primes. This formula is fun to prove: just use the geometric series to expand each factor, multiply them out and see what happens! Using this, Riemann and von Mangoldt derived an explicit formula for how many primes are less than a given number as a sum over the *zeros* of the Riemann zeta function. Instead of showing you this formula, I'll just urge you to watch a *movie* of how it works: 1) Matthew Watkins, Animation: the prime counting function pi(x), http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm Thanks to this formula, information about the Riemann zeta function is secretly information about the distribution of primes! For example, the Riemann Hypothesis says that when we analytically continue the zeta function to the complex plane, the only zeros occur at negative even integers and numbers with real part equal to 1/2. And, knowing this would be equivalent to knowing that the number of primes less than x differs from Li(x) = integral_2^x dt/ln t by less than some constant times sqrt(x) ln(x). Everyone feels sure these facts are true. But, despite over a century of hard work and a milliondollar prize offered by the Clay Mathematics Institute, nobody has come close to proving them! It's known that apart from the negative even integers, the only place the Riemann zeta function can vanish is in the strip where 0 < Re(s) < 1 But, nobody has been able to show that all the zeros in this "critical strip" lie on the line Re(s) = 1/2 Of course, this can be checked in special cases. The current record may belong to Xavier Gourdon, who on October 12th of 2004 claimed to have shown  with the help of a computer  that the first TEN TRILLION zeros in the critical strip lie on the line Re(s) = 1/2. 2) Xavier Gourdon, Computation of zeros of the Riemann zeta function, http://numbers.computation.free.fr/C...oscompute.html Alas, such monster calculations don't seem helpful for proving the Riemann hypothesis. They're more useful when it comes to formulating and testing conjectures about the *statistical properties* of the zeros. The most famous of these traces its way back to a teatime conversation between Hugh Montgomery and Freeman Dyson... you can read the story here: 3) K. Sabbagh, Dr. Riemann's Zeros, Atlantic Books, 2002, pp. 134136. Story about Hugh Montgomery and Freeman Dyson also available at http://www.maths.ex.ac.uk/~mwatkins/zeta/dyson.htm The upshot is that the distribution of spacings between Riemann zeros closely mimics the spacings between eigenvalues of a large randomly chosen selfadjoint matrix. This suggests fascinating relations between the Riemann zeta function and quantum physics. In fact, one popular dream for proving the Riemann zeta function is to find a chaotic classical system whose quantized version has energy levels related to the Riemann zeta zeros! I would like to understand this stuff, but it all seems a bit indimidating  especially since the coolest aspects are the ones *nobody* understands. Luckily, the Riemann zeta function has spawned a lot of other functions called zeta functions and Lfunctions, and many of these are *simpler* than the original one  or at least raise fascinating questions that are easier to solve. Many of these are listed here: 4) Matthew R. Watkins, A directory of all known zeta functions, http://www.maths.ex.ac.uk/~mwatkins/...afunctions.htm Matthew R. Watkins, A directory of all known Lfunctions, http://www.maths.ex.ac.uk/~mwatkins/...functions.htm Lately I've been talking about zeta functions with James Dolan and also Darin Brown, whose work I mentioned last week. I feel some things are starting to make sense, so I'd like to explain them before it turns out they don't. I'll just list some zeta functions, so you can see what we're dealing with: A) The zeta function of a number field. A "number field" is something you get by taking the rational numbers and throwing in some algebraic numbers. One can define "algebraic integers" for any number field, and they act a lot like the ordinary integers. So, one can define a zeta function for any number field. Technically, we do this by summing over all nonzero ideals I in our number field: zeta(s) = sum_I I^{s} where I is a number called the "norm" of I. We also have an Euler product formula: zeta(s) = product_P 1/(1  P^{s}) where we take the product over prime ideals. For example, if our number field is the rational numbers, its algebraic integers are the ordinary integers. So, each ideal consists of multiples of some number n = 1,2,3,..., and its norm is just n, so we get: zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + .... A more fun example is the number field Q[i], where we take the rational numbers and throw in a square root of 1. Here the algebraic integers are the socalled "Gaussian integers" Z[i], namely guys like a + bi where a and b are ordinary integers. In this example it's easiest to work out the zeta function using the Euler product formula. If you ask one of your number theory pals about prime ideals in the Gaussian integers, they'll say: "Well, the Gaussian integers are a principal ideal domain, so every ideal is generated by a single element. So, we can actually talk about *prime numbers* in the Gaussian integers. And there are 3 cases: INERT: An ordinary prime number of the form 4n+3 is also prime in the Gaussian integers: for example, 3. SPLIT: An ordinary prime numbers of the form 4n+1 is the product of two complex conjugate primes in the Gaussian integers: for example, 5 = (2+i)(2i). RAMIFIED: The ordinary prime 2 equals (1+i)(1i), but here the two factors give the same prime ideal, since (1i) = i (1+i), and i is invertible in the Gaussian integers. The three cases are called "inert", "split", and "ramified", and we get all primes in the Gaussian integers this way." So, the zeta function of the Gaussian integers goes like this: zeta(s) = 1/(1  2^{s}) 1/(1  3^{2s}) 1/(1  5^{s}) 1/(1  5^{s}) ... I went just far enough to show you what happens for each kind of prime. As you might expect, we get two factors for each prime that splits. I should explain the other details, but number theory is best absorbed in small doses, especially if you're a physicist. The main lesson to take home is this: A number field is like a funny sort of "branched covering space" of the set of ordinary primes. Sitting over each ordinary prime there are one or more prime ideals in our number field: 2+i 3+2i  1+i  3   7  11   GAUSSIAN INTEGERS 2i 32i 23571113 INTEGERS And, the zeta function records the details of how this works! For more on this covering space philosophy see "week205" and "week213". This geometrical metaphor lies behind a lot of the really cool work on number theory. B) The zeta function of a discrete dynamical system. A "discrete dynamical system" consists of a set X together with a onetoone and onto function f: X > X Here X is the set of "states" of some physical system, and f describes one step of time evolution. For each integer n we get a function f^n: X > X Since the integers are called Z, mathematicians would call our discrete dynamical system a "Zset". Whatever you call it, its zeta function is defined to be: zeta(s) = product_P 1/(1  P^{s}) where P ranges over all periodic orbits and P is the *exponential* of the size of this periodic orbit. This is like an Euler product formula with the periodic orbits being the "primes". Just as every natural number can be uniquely factored into primes, every discrete dynamical system can be uniquely written as a disjoint union of periodic orbits. This explains the exponential in the definition of P above: primes like to multiply, while sizes of orbits like to add. One nice thing about this zeta function is that when we take the disjoint union of two discrete dynamical systems, their zeta functions multiply. Another nice thing is that the zeta function of a dynamical system completely describes it up to isomorphism, at least when the set X is finite. Decategorification at work! We can also rewrite this zeta function as a sum: zeta(s) = sum_I I^{s} where I ranges over all formal products of periodic orbits, and we define P_1 ... P_n = P_1 ... P_n. Even better, examples A) and B) overlap. I'll explain how later, but the key is to associate to any number field and any prime p a discrete dynamical system f: X > X called the "Frobenius automorphism". This gives a zeta function. It works best if we take the exponential of the size of each periodic orbit using the base p instead of base e. Then, if we multiply all these zeta functions, one for each ordinary prime p, we get the zeta function of our number field! C) The zeta function of a continuous dynamical system. Now suppose X is some topological space and we have a time evolution map f_t: X > X for each real number t. We can define a zeta function zeta(s) = product_P 1/(1  P^{s}) where P ranges over all periodic orbits and P is the exponential of the "period" of P  meaning the time it takes for points on this orbit to loop around back to where they started. A famous example is when we have a Riemannian manifold. A free particle moving around on such a space will trace out geodesics, giving us a dynamical system. The analogue of primes are now "prime geodesics": periodic geodesics that loop around just once. The "covering space" philosophy described in example A) can now be taken literally! If the Riemannian manifold M' is a covering space of M, any prime geodesic P in M defines a deck transformation of M'. This transformation acts on the set X of prime geodesics sitting over P, so we get a onetoone and onto map f: X > X This is exactly like the "Frobenius automorphism" in number theory! All this is particularly interesting when our manifold is a quotient of the upper halfplane by a discrete group  see "week215" for more on this. The reason is that some of these quotients are related to number theory. So, we get some direct interactions with example A). D) The zeta function of a graph. We can take the idea of "periodic geodesic on a Riemannian manifold" and vastly simplify it by looking at closed loops in a graph with finitely many edges and vertices. We get a zeta function zeta(s) = product_P 1/(1  P^{s}) where P ranges over all "prime loops" in our graph: loops that don't backtrack or loop around more than once. Now P is the exponential of the length of the loop. The "covering space" philosophy still applies, since we can define what it means for a graph G' to be a covering space of a graph G. Any prime loop P in G defines a deck transformation of G'. This acts on the set X of prime loops sitting over P, so we get a onetoone and onto map f: X > X which again deserves to be called the "Frobenius automorphism". E) The zeta function of an affine scheme. Given a commutative ring, we can think of it as the ring of functions on some space. The zeta function of the commutative ring then counts the points of this space. To make this precise, we cleverly invent a kind of space called an "affine scheme", which is secretly JUST ANOTHER NAME FOR A COMMUTATIVE RING. So, any commutative ring R gives an affine scheme called Spec(R), purely by our fiendish definition. If we take a function and evaluate it at a point, we should get a number. This should give a homomorphism from functions to numbers. But in algebraic geometry, "numbers" can be elements of any field k. So, let's say the "kpoints" of Spec(R) are the homomorphisms from R to k. (This is a bit nontraditional, but I really need this here. For a more traditional alternative, see "week205".) In particular, for any prime p we can take k to be the algebraic closure of the field with p elements. Let X be the set of kpoints of some affine scheme Spec(R). Then comes something wonderful: if x is a kpoint, so is x^p, since "raising to the pth power" is an automorphism of k. So, we get a map f: X > X sending x to x^p. This is called the "Frobenius automorphism"! Since f is a discrete dynamical system, we can define its zeta function as in example B): zeta(s) = product_P 1/(1  P^{s}) where P ranges over all periodic orbits, and P is the exponential of the size of the periodic orbit, defined using the base p. So far, this the zeta function of our affine scheme "localized at p". If we multiply a bunch of factors like this, one for each ordinary prime p, we get the zeta function of our affine scheme. If you know about schemes that aren't affine  like projective varieties, such as elliptic curves and other curves  you'll see this definition works for them too. If you know someone else's definition of the zeta function of a scheme, it may not look like what I wrote here! But don't panic. The reason is that people like to express the zeta function of a discrete dynamical system f: X > X in terms of the number of fixed points of f^n. When f is the Frobenius automorphism, these are usually called "points defined over the field with p^n elements". So, you'll see lots of formulas for zeta functions phrased in terms of these.... Okay. Enough examples. There are a lot more, but I think these are the simplest. I hope you see how all these examples are just different expressions of the same idea. To go further, I would tell you how there are versions of the Riemann Hypothesis in all these examples, and also things called "Lfunctions", and lots of theorems and conjectures concerning them, too! It's a wonderful example of the unity of mathematics. But, it's also a wonderful example of how this unity is obscured by people who zoom in on their own favorite special case and its particular technical complexities while never discussing the big picture. You wouldn't believe how hard it's been for me to figure out what I just told you! It's like trying to learn English by reading the US legal code, or learning basic chord progressions by listening to Schoenberg. If you're just trying to get started, here's one of the more readable introductions: 5) Anton Deitman, Panorama of zeta functions, available as math.NT/0210060. Audrey Terras has a lot of nice slide presentations about the zeta functions and Lfunctions of graphs: 6) Audrey Terras, Artin Lfunctions of graph coverings, available at http://math.ucsd.edu/~aterras/artin1.pdf More on Lfunctions, available at http://math.ucsd.edu/~aterras/artin2.pdf Here's a paper written in broken English but making a very serious attempt to explain things to the nonexpert: 7) David Zywina, The zeta function of a graph, available at http://math.berkeley.edu/~zywina/zeta.pdf He gives a characterization of graphs whose zeta functions satisfy an analogue of the Riemann Hypothesis. Strangely, this analogue involves *poles* of the zeta function in the critical strip 0 < Re(s) < 1 Is this a real difference or the result of some difference in conventions? Finally, I should explain some more technical stuff about zeta functions and fixed points, just so I don't forget it. Suppose we have a discrete dynamical system f: X > X and let fix(f^n) be the number of fixed points of the nth iterate of f. We can define a weird function like this: Z(u) = exp(sum_{n>0} fix(f^n) u^n / n) You'll often see formulas like this running around, especially when f is some sort of "Frobenius automorphism". Sometimes people even call these guys zeta functions. But what in the world do they have to do with zeta functions??? Let's see. Suppose first that X consists of a single cycle of length k. Then f^n has k fixed points when n is a multiple of k, but none otherwise, so: Z(u) = exp( ku^k/k + ku^{2k}/2k + ku^{3k}/3k + ... ) = exp( u^k + u^{2k}/2 + u^{3k}/3 + ... ) = exp( ln(1/(1  u^k)) ) = 1/(1  u^k) This is starting to look more like the zeta functions we know and love. It looks even better if we pick some prime number p and define zeta(s) = Z(p^{s}) Then we get zeta(s) = 1/(1  p^{ks}) which is precisely what we'd get using the definition in example B). Furthermore, for both that old definition and our new one, the zeta function of a disjoint union of discrete dynamical systems is the *product* of the zeta functions of the parts. Since every discrete dynamical system is a disjoint union of cycles, we conclude that the definitions *always* agree: zeta(s) = Z(p^{s}) with Z(u) = exp(sum_{n>0} fix(f^n) u^n / n) is always equal to the zeta function defined in example B). So, don't let anyone fool you  there aren't lots of completely different kinds zeta functions! There's just a few kinds, and we could probably boil them down to just ONE kind with some work. Quote of the week: "Some decades ago I made  somewhat in jest  the suggestion that one should get accepted a nonproliferation treaty of zeta functions. There was becoming such an overwhelming variety of these objects."  Atle Selberg  Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumpingoff point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html 


#2
Oct1106, 03:02 PM

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Zeta functions are wondrous beasts! Thanks for this awesome issue of
TWF. John Baez wrote: > May 23, 2005 > This Week's Finds in Mathematical Physics  Week 216 > John Baez > [...] > E) The zeta function of an affine scheme. Given a commutative ring, > we can think of it as the ring of functions on some space. The zeta > function of the commutative ring then counts the points of this space. I realize this wasn't your focus but I feel compelled to mention the Weil conjectures in connection with this. In the related case of an algebraic variety V over a finite field F_q (where q = p^n for some prime p) we have the exponential generating function Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!) which is the socalled local zeta function of the variety. Presumably the word "local" is used because the F_{q^n} are in some sense increasingly local versions of F_qso the local zeta function counts points of V over increasingly local fields. Part of what Weil conjectured was that Z(T) should always be a rational function. In fact he conjectured a pretty specific form. All this is extremely unexpected! It turns out that it would follow from a Lefschetzstyle fixed point formula for a sufficiently rich cohomology theory. But the problem at the time was no such cohomology theory was knownthe only natural topology on V is the Zariski topology, which is very pathological. So all the usual ways of setting up cohomology failed to give much useful information. Grothendieck eventually figured out a way around these problems by inventing etale cohomology. Deligne used this to eventually prove the Weil conjectures. I've been trying to learn some of this stuff from J. Milne's lecture notes on etale cohomology at http://www.jmilne.org/math/CourseNotes/math732.html which are excellent but very heavy going (unsurprisingly). The first chapter or two are relatively chattythey are definitely worth checking out if you are interested in a vague impression of what some of this stuff is about. Per 


#3
Oct1106, 03:02 PM

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Dr. Baez et al...
Where's the physics application in this post? I read it twice and I'm uncertain how it is applicable to physics, help a dense guy:). Ken [Moderator's note: Entire bottomquoted post deleted. Please quote as much of the post as necessary, but as little as possible, and add your reply after that. P.H.] 


#4
Oct1106, 03:03 PM

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This Week's Finds in Mathematical Physics (Week 216)
In article <1117079187.198693.76220@z14g2000cwz.googlegroups.com>,
Per Vognsen <per.vognsen@gmail.com> wrote: >Zeta functions are wondrous beasts! Thanks for this awesome issue of >TWF. You're welcome! I plan to write more about this stuff... though I know from experience that as soon as I *plan* to write about something in This Week's Finds, I decide it's a chore and switch to writing about something else! >I realize this wasn't your focus but I feel compelled to mention the >Weil conjectures in connection with this. Excellent! These were indeed on my mind... >In the related case of an >algebraic variety V over a finite field F_q (where q = p^n for some >prime p) we have the exponential generating function > >Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!) > >which is the socalled local zeta function of the variety. Right. Part of what used to bug me immensely is that I didn't see how this formula was related to other things called zeta functions. That's why I'm so happy to learn about a very simple idea  the zeta function of a discrete dynamical system  and to realize that the above zeta function comes from a dynamical system where the time evolution map is called the "Frobenius". I'm sure you understand this already, but just for everyone else: this is why "week216" I gave a proof that for any discrete dynamical system f: X > X we can define a function generalizing the one you wrote down: Z(T) = exp(sum_{n>0} fix(f^n) T^n / n) and then zeta(s) = Z(p^{s}) equals what I'd consider the "usual" zeta function of a dynamical system, namely zeta(s) = product_P 1/(1  P^{s}) where P ranges over all periodic orbits, and P is the exponential of the size of the periodic orbit, defined using the base p. >Presumably the word "local" is used because the F_{q^n} are in some sense >increasingly local versions of F_qso the local zeta function counts >points of V over increasingly local fields. Right. The fields F_{q^n} are "local fields" in a precise technical sense, while the rational numbers are a "global field"  but this precise technical sense is based on a simple beautiful idea, namely that the rational numbers are functions on a space whose points are just the prime numbers (together with the "real prime"). So, keeping track of the power of a prime that shows up in a rational number is a way of studying that number "locally". And, just as global problems in geometry get easier if we first tackle them locally, the same is true of all problems in number theory! >Part of what Weil conjectured was that Z(T) should always be a rational >function. In fact he conjectured a pretty specific form. All this is >extremely unexpected! It was less unexpected to him than to most of us, because he knew that there was a very simple formula relating the cohomology of complex Grassmannians to the number of points of these Grassmannians over finite fields. I talked about this formula at enormous length in "week187" and previous weeks. I knew it was related to Weil's conjectures. But, only recently did I learn that Weil was motivated by precisely this (and other facts) to make his conjectures! 


#5
Oct1106, 03:04 PM

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In article <d764b9$kke$1@glue.ucr.edu>, baez@galaxy.ucr.edu (John Baez)
wrote: > In article <1117079187.198693.76220@z14g2000cwz.googlegroups.com>, > Per Vognsen <per.vognsen@gmail.com> wrote: > > >In the related case of an > >algebraic variety V over a finite field F_q (where q = p^n for some > >prime p) we have the exponential generating function > > > >Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!) > > > >which is the socalled local zeta function of the variety. > > Right. Part of what used to bug me immensely is that I didn't > see how this formula was related to other things called zeta functions. (I think there shouldn't be a factorial there) It's sort of weird, I think, because for a lot of application it seems like the Lfunction is much more analogous to the Riemann zeta function. Of course, Hartshorne (Ex C.5.4) has the following (paraphrased) Given a Scheme X of finite type over Spec Z, let Zeta_X(s) = \prod (1  N(x)^s)^1 where the product is taken over all closed points x and N(x) is the number of elements of the residue field k(x)= O_x / m_x. Doing this for X = Spec Z gives you the usual zeta function. Then, if X is of finite type over F_q, we can do the same thing and get Zeta_X(s) = Z(X,q^s) To define Z(X,t), let N_r be the number of rational points of X x_F_Q \bar{F_q} (the algebraic closure of F_q) over F_q^r. Then stick these into Z(X,t) = Exp(\sum N_r t^r/r) When I'm more awake I might actually try to understand this and relate this definition to the one your state (N_r is also the number of fixed points of the Frobenius automorphism). Aaron 


#6
Oct1106, 03:04 PM

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In article <slrnd95p5l.8r3.robert@localhost.localdomain>,
Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: >will there be a part II of this post where you explain the relation to >Gutzwiller's trace formula and zeta functions of operators used to >regularize QFT's? Umm, right now I'm much more fascinated by other things, like trying to understand the network of ideas related to zeta functions and Lfunctions in number theory  stuff like the Weil hypotheses, the TaniyamaShimuraWeil conjecture, and the Langlands program. After about a year of battering my head against these, they seem like they're starting to make sense... either that or I've knocked myself silly! I want to write more about this stuff. So, I can't promise to get around to the other aspects of zeta functions that you mention. But, they *are related*, so I should keep them in mind. 


#7
Oct1106, 03:04 PM

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In article <abergman14F5A4.02463127052005@localhost>,
Aaron Bergman <abergman@physics.utexas.edu> wrote: >When I'm more awake I might actually try to understand this and relate >this definition to the one you state (N_r is also the number of fixed >points of the Frobenius automorphism). I think I understand this stuff now; the key is that it has absolutely nothing to do with algebraic geometry, schemes, Frobenius automorphisms or all that jazz. It's just a fact about a 11 and onto map f: X > X We can define zeta(s) = product_P 1/(1  p^{Ps}) where P ranges over all periodic orbits of f and P is the size of the periodic orbit. We can also define Z(u) = exp(sum_{n>0} fix(f^n) u^n / n) where fix(f^n) is the number of fixed points of f^n. Then we have zeta(s) = Z(p^{s}) I gave the proof in "week216" (using slightly different notation). We can then apply this to algebraic geometry by letting f be a Frobenius automorphism, and see that two differentlooking definitions of the "zeta function" of a scheme agree. But, we can also apply it to dynamical systems and all sorts of other stuff.  Michael Berry on the dreamtof "Riemann dynamical system". He is: "... absolutely sure that ... someone will find a clever way to make it in the lab. Then you'll get the Riemann zeros out just by observing its spectrum." "Finding this system could be the discovery of the century." Berry says. It would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos. Until [it is found] we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It is a tantalising thought." From: Martin Gutzwiller, Quantum Chaos, Scientific American, January 1992, available at http://www.maths.ex.ac.uk/~mwatkins/...ntumchaos.html 


#8
Oct1106, 03:05 PM

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baez@math.removethis.ucr.andthis.edu (John Baez) writes:
> Umm, right now I'm much more fascinated by other things, like trying > to understand the network of ideas related to zeta functions and > Lfunctions in number theory  stuff like the Weil hypotheses, the > TaniyamaShimuraWeil conjecture, and the Langlands program. After > about a year of battering my head against these, they seem like > they're starting to make sense... either that or I've knocked myself silly! > I want to write more about this stuff. So, I can't promise to get around > to the other aspects of zeta functions that you mention. But, they *are > related*, so I should keep them in mind. When I was a graduate student and a while afterwards, I wanted to be very sure of never doing war related research. After getting involved with number theory and algebraic geometry and mathematical logic, I thought I had found remote corners of mathematics without discernible military applications. Then I found out that number theory is important in code breaking and that algebraic geometry and number theory are of increasing importance in physics. I still haven't made up my mind whether the physics they are important in is really physics or is really mathematics, but I'm a lot less comfortable in my useless niche than I used to be. I certainly want to know the answers to basic questions, both in mathematics and in physics, but I'm also concerned with the fact that there is always the potential to use it to bring ever greater forms of devastation into existence and I don't want to be a part of that. I'm not really sure how to deal with this renewed concern or how to really measure the relevance of any particular area of research. I'd be interested in knowing how others deal with these issues. I guess that what started me wondering about this again was when the physicists at IHES wanted to have a copy of the preliminary manuscript for my book, Moduli of Abelian Varieties, which I had been happy to think had nothing to do with reality and was just endless fun with purely mathematical objects.  Ignorantly, Allan Adler <ara@zurich.csail.mit.edu> * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not reflect in any way on MIT. Also, I am nowhere near Boston. 


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