Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems


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Sign In Register Help Cart. Cart items. Toggle navigation. Search Results Results 1 -4 of 4. CreateSpace Independent Publishing Platform. Used - Very Good. Realizing the difficulty of negotiating with a dead guy and the absurdity of having a function that could actually be many different functions, Dirichlet decided that he should rename his function and add on some sort of marker to indicate which n he was using.


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  • As such, he started using L instead of Z, and he began to call his functions L s, n , which means the L function where the pattern in the numerators repeats every n terms. And the Greek letter chi, pronounced kai takes care of the mathematicians compulsive need. Well, there are a couple of instances where the answer is yes.

    For instance, theres this one:. Theres also this one:.

    Trolling Euclid An Irreverent Guide to Nine of Mathematics Most Important Problems

    Lets be honest, though; when we say, Does this do anything cool? In other words, were looking to see if something like the Riemann Hypothesis shows up here, too. Well, when we asked that question last time, we immediately ran into the problem where the function started blowing up. For every choice of n, L s, m blows up for s 0. Some of them even blow up for s 1. Well have to address that first.

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    While Johann Peter Gustav Lejeune Herbert Walker Iodine Shaquala Dumbledorf Hussein Dirichlet could well have used his new function to generalize Riemanns Hypothesis, he declined to do so, citing the fact that he was, quote, already dead by the time that Riemann had formulated his hypothesis. Because of this act of cowardice on Dirichlets part, it was left to another pair of mathematicians, the infamous tag team of Adolf Hurwitz and Adolf Piltz or The Amazing Adolfs, as they were known to fans at the time , to finally generalize the Riemann Hypothesis so that these Dirichlet functions would be included, too.

    Hurwitz and Piltz were an interesting pair. Hurwitz was the brainier of the two, as he was particularly facile with all different types of number theory and analysis and, as a result, has several important functions named after him today. Piltz, by contrast, was more of the brawn behind the operation, often overpowering opponents before forcing them into submission with his patented Complex Integration finishing move.

    Together, they proved to be an unstoppable force in the math world; in , they would go on to win the Mathematics Tag Team Championship Belt from fellow mathematicians David Hilbert and Herman Minkowski1 when, in one of the most famous tag team matches in mathematics history, Hurwitz discovered a new automorphism theorem for surfaces with genus larger than one and Piltz kicked Minkowski in the face2. It was Hurwitz, the smarter of the pair, who found these L s, n s that Dirichlet had unleashed, and, much as Riemann before him, took them into his office and tried to get them to stop blowing up.

    After much work and many failed attempts, Hurwitz finally discovered his function that would act as a patch; it would act like L s, n everywhere that the L s, n was defined, but it wouldnt be nearly as temperamental. Unfortunately, Hurwitz was so excited by his discovery of a patch that he forgot to come up with another name for it, so we now call this patched function L s, n which, as you might confusedly remember, is the same name as the function that Hurwitz was trying to patch in the first place.

    Here, by possibly, we mean that it depends which function L s, n youve chosen. For some choices, L s, n is defined for all s, while for others, its defined for all s except 1. Minkowski and Hilbert were, of course, known to fans at the time as Herm and the Hat. Shut up, readers. My storys better. Piltz center, facing camera and Hurwitz left discuss methods of analytic continuation of L-functions with colleagues.

    Despite Hurwitzs extraordinary efforts in discovering these patch functions, we mathematicians have decided to call these things Dirichlet Lfunctions on the grounds that Dirichlet has more syllables and therefore sounds more impressive to say. Hurwitz may not like it, but thats his problem.

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    Now that we have our functions that are defined for every s and act like the Ls that Dirichlet was thinking about, we can finally ask when they hit zero. This was the question that faced the Adolfs in Having done his share of the work, Hurwitz now turned things over to Piltz, the brawnier half of the duo, who would attempt to finish the problem using one of his famed finishing moves.

    Unfortunately, Piltz was a bit too strong for his own good, and L s, n was only able to respond, Mmfghh wmmph thfffff This obviously wasnt much help, so Piltz was forced to make an educated guess. He first noticed that some of the zeroes were pretty obvious:. Figuring out which L s, n corresponds to which of these solutions isnt really hard, so those zeroes are pretty well taken care of. Now, we get to the harder part: other than the easy answers above, where are these functions zero? Piltz thought long and hard about this problem, looking for all manner of leads and clues and even trying some smelling salts in an attempt to wake L s, n up so he could put it in a chokehold again.

    Now, on the one hand, we should be disappointed in Piltz for copying so flagrantly. On the other hand, it looks, miraculously enough, like his conjecture is probably right. In other words, what happens in Riemanns case happens in every case, no matter which L s, n you choose; the interesting zeroes are always in the same place.

    Piltz stated his new conjecture in and demanded that it be named Piltzs Conjecture. However, the mathematical community, although impressed by the simplicity and elegance of the conjecture, were not big fans of Piltz particularly as many of them had previously been on the wrong end of Piltzs strong-armed tactics , and thus were in no mood to accede to his demands. As a result, the math community decided that the conjecture should instead be called the Generalized Riemann Hypothesis 3 : Generalized Riemann Hypothesis: Pick an L s, n.

    It doesnt matter which. Believe it or not, the grudge against Piltz has continued to this day. Think Im kidding? According to the American Institute of Mathematics, one of the most important math organizations in the world, GRH is occasionally called Piltz conjecture, but the conjecture of a Riemann Hypothesis for Dirichlet L-functions is generally viewed as obvious generalization which should not be attributed to a particular person.

    Someones bitter. As important as the Riemann Hypothesis is, the generalized version is probably even more important because if it were true, it would imply many, many important results. Well talk about some of these in the next couple of sections. The reason that these two statements arent at odds with each other is because mathematicians believe that whatever tools and techniques finally prove the original version of the Riemann Hypothesis will probably give us a lot of insight into how to prove and may, in fact, actually prove GRH.

    This is why the original version is the million dollar problem; its widely seen as the first domino that will make all of the Riemann Hypotheses fall. If youre keeping score at home, this means that in addition to making conjectures about mathematics, mathematicians are now also making conjectures about the proofs of these conjectures and how these conjectured proofs of conjectures will help prove other conjectures.

    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems
    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems
    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems
    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems
    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems
    Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems Trolling Euclid: An Irreverent Guide to Nine of Mathematics Most Important Problems

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