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William Stein -- Talk for Mathematics is a long conversation: a celebration of Barry Mazur
\documentclass{article}1\voffset=-1.6in2\textheight=1.41\textheight3\hoffset=-.75in4\textwidth=1.1\textwidth5\usepackage{url}6\usepackage{tikz}7\usepackage{hyperref}8\usetikzlibrary{matrix,arrows}9\include{macros}10\DeclareMathOperator{\sss}{ss}11\DeclareMathOperator{\Li}{Li}12\renewcommand{\ss}{\sss}13\renewcommand{\b}{\mathfrak{b}}14\renewcommand{\c}{\mathfrak{c}}15\newcommand{\cN}{\mathcal{N}}16\title{RH Notebook}17\author{William Stein}18\date{}19\begin{document}20\maketitle21\tableofcontents2223\section{August 18, 2009: Getting going}2425I read the first 35 pages of Rockmore's book. It is like26fingernails on a chalkboard.2728RH book todo:29\begin{itemize}30\item make a wiki page?31\item get my hg repo up to speed32\item read through text making list33\end{itemize}3435Here is a todo list while reading the book:36\begin{enumerate}37\item Make a list of the books --both popular and not-- about38RH. (mentioned on page 1). I have Patterson (serious) and Edwards39(serious) on my desk, and Sabbagh (popular) and Rockmore40(cringe-inducing) on my desk too. There could be other popular41books that have chapters about RH that are good (or not). E.g.,42``The Millennium Problems'' by Keith Devlin has chapter 1 about RH43(the BSD chapter of that book sucks, but I maybe the RH chapter is44good?).45\item We say ``least mathematical background required'' but having46tested our booklet on students, I would say that we do not succeed47there. We could make our booklet 2 times as long and require less48math background. I've slowly come to think this would be worth it.49And we could make it much longer still by adding way more50illustrations (generated by Sage) and lots of prose explaining what51is in the illustrations (little guided tours), and this would also52be worth it.5354\item I definitely want to say more in the book about how RH informs55complexity analysis in computational number theory. This is perhaps56the main way RH appears in modern computational number theory.57Maybe there are some very simple down-to-earth examples of this58principle at work.5960\item I would almost like to restructure things so the illustrations61are much more extensive and integral and included in the main text.62Then the additional Sage interacts are merely an ``additional63resource'' for those wishing to investigate further. They're an64added bonus. But they can also be safely ignored.6566\item Typo: ``websitte''.6768\item I think we should remove the business about how long it should69take to read the book. Let the reader start reading and decide for70themselves. Otherwise, they might feel insecure and wonder all the71time if they are taking ``too long''.7273\item ``less than 100, 10,000, 1,000,000, `` that looks at first74glance like a single huge number. Maybe make it three statements.75less than 100? less than 10,000? less than 1,000,000?7677\item Picture of Bott?7879\item Picture of Zagier?8081\item This sentence: ``If we are to believe Aristotle, the early82Pythagoreans thought that the principles governing Number are “the83principles of all things,” the elements of number being more basic84than the Empedoclean physical elements earth, air, fire, water.''85I've been looking at other popular math books, and they never just86assume the reader knows who Aristotle is, Pythagoreans were, or what87Empedoclean means. In fact, I have no idea what Empedoclean means,88and I can easily forgot {\em when} the Pythagoreans were around.89That said, I would rather say nothing to say something wrong.9091\item Descarte picture? Are there any? [[Yes -- see Wikipedia]]9293\item Speaking of ``wrong'' (see above), somebody emailed me this:94\begin{verbatim}95In a text "Elementary Number Theory" in section 7.1.2, you have96an implementation of the sieve of Eratosthense. Melissa O'Neill97wrote a paper, "The Genuine Sieve of Eratosthenese". I do not98believe that your program meets here criteria for being the99genuine sieve of Eratosthenses. I used IDLE on an IBM/PC to100run your program and crashed, if I entered the value of 200000.101I can create a list of primes at least up to 200000 if I use a102Python program that meets her criteria.103\end{verbatim}104We should read Melissa O'Neill's paper to see what the deal is.105106\item ``Contemporary physicists dream of a “final theory.”'' Do they107really? In what sense?108109\item ``Don Quixote encountered this...'': Who is he exactly? A110fictional character, a person? When? I've heard of him, but111honestly I've never read anything nontrivial about him, and I doubt112most of our readers will have. They might see him mainly as a113mysterious person whose last name is hard to pronounce.114115\item Why exactly do Cicada's come out every 17 years? I saw Bruce116Jordan in Princeton recently and we started talking about this117(since they have Cicada's there), and I quickly realized I didn't118really have a clue.119120\item ``Philolaus (a predecessor of Plato)'' that isn't a good enough121introduction to Philolaus, given that it is the first mention of122Plato. Again, many readers might not know Plato so well. Heck, I123don't. I view all the above remarks as opportunities to expand our124book's readership and mission a bit, rather than criticisms of it.125126\item ``But, until Euclid, prime numbers seem not to have been singled127out as the extraordinary math- ematical concept, central to any deep128understanding of numerical phenomena, that they are now understood129to be.'' Here we are foreshadowing Euclid's proof that there are130infinitely many primes, etc. But this is also the first time Euclid131is even mentioned. To a casual reader it just feels that it's a132point in an outline that hasn't been filled in.133134\item Instead of starting with the 300 factoring example, perhaps we135should first start with a smaller one where we can list {\em every}136single factorization tree/order. This makes things feel less {\em137abstract}, since the reader doesn't have to imagine all the138missing factorizations.139140\item ``more than 100 digits, to your computing machine and ask it to141multiply them together: you will get their product N = P × Q with142its 200 or so digits in a few microseconds.'' I just checked and it143is a few hundred {\em nanoseconds} to do that. So lets change to144``few hundred nanoseconds'' or perhaps better ``a just under a145microsecond.''146147148\item Our proof of the infinitude of primes on page 8 is the first149time in the book we use symbolic notation, give a proof, reason150abstractly, etc. I wonder if we could do a little more to prepare151the reader. I just read Rockmore's horrendous proof of the same152thing in his book on RH -- it's pages of tedium to say in words what153takes 1 second with symbols. But I'm attracted to the challenge of154doing something a little bit in between, e.g., having an example.155156\item Move our discussion of EFF cash prize up, since the prize was157just won! I wonder if there are any press releases about the prize158being awareded, which we could cite or point to?159160\item Here is Sage actually computing the decimal digits of the161biggest known Mersenne prime:162\begin{verbatim}163sage: time a =2^43112609-1164CPU times: user 0.01 s, sys: 0.01 s, total: 0.01 s165Wall time: 0.02 s166sage: time s = str(a)167CPU times: user 12.23 s, sys: 0.99 s, total: 13.22 s168Wall time: 13.63 s169sage: s[-10:]170'6697152511'171sage: time sum(a.digits())172CPU times: user 15.25 s, sys: 1.07 s, total: 16.33 s173Wall time: 16.84 s17458416637175\end{verbatim}176177\item ``But there is no obvious way'' -- maybe ``no known way''?178179\item ``In Figure 3.3 we use the primes 2, 3, 5, and 7 to sieve out180the primes up to 100, where instead of crossing out multiples we181grey them out, and instead of circling primes we color their box182red.'' I could make a sequence of figures where we do cross them183out too? The grey background is hard to see and probably hard to184print, so I can do better there too.185186\item For all these questions: ``Are there infinitely many pairs of187primes whose difference is 4? Answer: equally unknown. Is every even188number greater than 2 a sum of two primes? Answer: unknown. Are189there infinitely many primes which are 1 more than a perfect square?190Answer: unknown.'' we could give precise references into Richard191Guy's book ``Unsolved Problems in Number theory'', which in turn has192a very good collection of references and more detailed description193of each problem. This would be a good endnote.194195\item I may as well draw a plot of $\text{Gap}_k(X)$ for various k together196on one plot.197198\item We have several natural {\em sections} already in the first 8199pages, but don't break them up as such. We should. It would make200things easier to navigate. We have a section ``what are primes''. Then ``prime gaps''. Then ``multiplicative parity''.201202\item ``Here is some data:''... and a weird big page break?203204\item On page 12, the references to Borwein etc. should of course be205moved to an endnote.206207\item On page 15 (Fig 5.4), it would be nice to have a less zoomed out208big figure for starters, to look at while reading along. Basically209like Fig 5.5, which looks very nice. Those figures could be bigger210too. I really like this part of the text though, where we are211spending a lot of care explaining the mathematics.212213\item I need to figure out how to be very precise in placing all the214figures where we want, not where latex wants. Right now there215placement significantly detracts from readability.216217\item ``The particular issue before us is, in our opinion, twofold,218both applied, and pure: can we curve-fit the “staircase of primes”219by a well approximating smooth curve?'' I think it would be worth220emphasizing that our smooth curve must be given for a ``formula''.221I mean, a typical reader might just think ``of course any kid could222take a pencil and draw a smooth curve through the staircase of223primes''. But to get a curve given by an sort of analytic formula224at all and which happens to have {\em anything } at all to do with225the function $\pi(X)$ -- well that seems really hard. A typical226reader might have no idea where to start to do that. Maybe we can227express this somehow?228229\item `` the chances that a number N is a prime is inversely230proportional to the number of digits of N''. Does there exist any231heuristic plausibility argument for this assumption that would make232sense to give at this point? Things are made a bit confusing since233the constant isn't 1, e.g., the probability that a number around a234billion is prime is not about ``1 in 9''.235236\item Pure and applied math. I think we should double the length of237this section by adding some examples. In particular, we could use238examples all of which will appear later! Examples of possible239illustrations include:240\begin{itemize}241\item something foreshadowing Fourier theory (applied)242\item random walks (finance?) (applied)243\item data compression (applied)244\item Mersenne primes, e.g., the Lucas-Lehmer test for primality (pure mathematics)245\item Goldbach's conjecture (pure)246\item The Hardy-Littlewood conjecture about asymptotics of $\text{Gap}_k(X)$. (pure)247\item Complex numbers (pure and applied); and with an endnote that248points at your book Imagining Numbers?249\end{itemize}250251\item Picture of Gauss (and I really like our Gauss dates)252253\item ``Roughly speaking, this means that the number of primes up to X254is X times the reciprocal of 2.3 times the number of digits of X .''255I think this is confusing to read. The reciprocal of 2.3 is kind of256funny, since 2.3 is already mysterious. It's really $1/log_e(10)$,257which is $0.43429448190325176...$, or basically $.4$. Maybe better258would be ``Very roughly speaking, this means that the number of259primes up to $X$ is about $X$ divided by twice the number of digits260of $X$.'' We can make a261table to illustrate this further, but also to emphasize that it's262not that close. Something like this:263\begin{verbatim}264sage: for i in [2..10]: print i, prime_pi(10^i), floor((10^i-1)/(i*2))265....:2662 25 242673 168 1662684 1229 12492695 9592 99992706 78498 833332717 664579 7142852728 5761455 62499992739 50847534 5555555527410 455052511 499999999275\end{verbatim}276277It's kind of convenient that for $99$, $999$, and $9999$, the278approximations got by taking ``$X$ divided by twice the number of279digits of $X$ are very close to $\pi(X)$. Anyway, rounding to $0.5$280instead of $0.43$ makes it really simple to describe.281282\item We might do something to warn our reader that if they see283``$\log(X)$'' they shouldn't run in fear and think ``holy crud, I284have no idea what log is and I never understand that in high285school,'' since we are about to explain it. I'm imagining say my286brother as reading this -- he literally probably hasn't seen log287once in a decade though he is good with numbers (running five288businesses in San Diego). He told me that when he sees a page with289a mathematical formula involving symbols he doesn't know, he'll just290block it out. So if we sneak ``log'' in to a sentence or two before291we use it in a formula, it'll get by that filter (which is probably292pretty common with non-math people).293294I'm imagining a solution like this:295\begin{itemize}296\item We figure out where logs first came from and give one sentence about297this (I think they arose in doing arithmetic efficiently?)298299\item We demystify log {\em before} using it in any formula by explaining300that it is ``about twice the number of digits''.301302\end{itemize}303304One other issue is that in much of math education, unfortunately $\ln$305means natural log and $\log$ means $\log$-base-10. It's really306annoying... We can mention this somehow.307308It might be worthwhile to remark that $e^x$ is the unique nonzero309function that equals its own derivative -- perhaps this is a way to310sneak in a mention of derivatives before later in the book where we311use them a lot more. Anyway there are two issues: (1) what are logs,312and (2) what is this ``natural log''?313314\item ``the 2004 US elections'' -- this will not be in people's minds315for a {\em book} so much. It may be better to remove or expand with316a statement about just how close they were with a reference. E.g.,317``the 2004 US elections, in which ... beat ... by a mere318... votes!'' Wasn't the 2000 election even closer, or am I319mis-remembering?320321\item ``So when Gauss thought his curve missed...'' let's compute the322square root explicitly here, i.e., just spell this out some more323(instead of leaving an exercise for the reader).324325\item ``devil fable'' I found this graphic via an image search on google:326327\url{http://blog.al.com/stantis/2007/11/Stantis-Devil%20in%20the%20Langford%20details.jpg}328329If the cover were modified, or the whole thing redrawn, it could be fun. The cover could say $primes up to X$. Or it could be replaced by the checklist...330331\item We write ``$\pi(X)$ for various large numbers $N$'' in our devil fable. Oops.332333\item We should draw an illustration of the checklist in the story.334It would be easy.335336\item We make the claim ``The average error (over-counted or337undercounted) would be proportional to $\sqrt{N}$.'' We do not338justify this claim at all. We might say that it follows from a339result about random walks. (Does it really follow from the central340limit theorem somehow?) Also, given that we assumed that the error341rate is 0.001\% can't we say what the constant in the proportion is?342Also, I think we could give an estimate of how far they would be off343for $N=3,000,000$. We could deduce Gauss's error rate, right?344345\item In figure 10.1 with plots of Li, pi, and X/Log(X), I should346put labels in the actual plot. It is lazy putting them only347in the caption.348349\item I should update the $X=4\cdot 10^{22}$ to whatever350the current record is, I think $10^{24}$, maybe. And also update351the reference, which may be wrong. Also, here is where we can possibly352discuss how to compute $\pi(X)$, or if not we can at least point353to an (extended) endnote. When this is done, be sure to search and354update all other references to $4\cdot 10^{22}$.355356\item We write -- `` an easier fact, which follows directly from357elementary calculus'' for the fact that $\Li(x)$ is asymptotic to358$X/\log(X)$. We should prove this rigorously in an endnote.359360\item ``It was proved in 1896 indepdently by Hadamard and de la Valle Poussin. ``361362(1) typo in ``indepdently''; we should say something about who these guys are,363and give links to Wikipedia (say).364365\item We write ``is much deeper than the Prime Number Theorem''. I366think the phrase ``is much deeper'' is mathematical jargon, because367popular math books would often have a little interlude to say368something about what deep means to mathematicians. It's basically369``difficult and any proof will use and influence a wide range of370mathematics''. So we too can add a little more to emphasize what we371mean by the word ``deep''. Or we can just say that the rest of this372paragraph explains what we mean (indeed, it does). Maybe everything373is perfect as is.374375\item ``It is the kind of conjecture that Frans Oort...'' let's have a376sentence about who Oort is. E.g., Dutch mathematician, born 19xx,377student of xxx... I might have a picture of him too.378379\item We write ``A proof of RH would, therefore, fall into the applied380category, given our discussion above.'' But we changed our381discussion above, so this is no longer quite true.382383\item I wonder if I could draw a 3d picture of an actual staircase384whose side profile is the plot of $\pi(X)$, but rendered at an angle385to look like a real staircase. This might be a nice illustration.386387\item In the section ``Tinkering with the carpentry of the staircase388of primes.'' I should draw several plots illustrating every single389one of the steps we discuss about tinkering with the staircase.390391\item ``These vertical dimensions might lead to a steeper ascent but392no great loss of information'' Maybe change to ``Since $\log(p)>1$,393these vertical dimensions lead to a steeper ascent but no great loss394of information.''395396\item ``Do not worry if you do not understand why our first and second397formulations of Riemann's Hypothesis are equivalent.'' We should398either rigorously prove this in an endnote (my preference at the399moment) or gave a reference that totally does it. I could imagine a400better student who has a more advanced background, who would benefit401by seeing a proof at this point. And it might help us keep things402straight... e.g., we had this equivalence wrong I think in some403version of our notes long ago.404405\item ``variety of equivalent ways we have to express Riemann’s406propose answers to the question'' -- I think ``propose'' should be407``proposed''.408409\item I'm worried that our second statement of RH is possibly410confusing because it says ``This new staircase is essentially square411root close''. However, given a line and curve the notion of close412is vague. What we really mean is that the function $\psi$ given by413the new staircase is an essentially square root approximation to the414function $f(x) = x$.415416\item Having just read ``Tinkering with the carpentry of the staircase417of primes.'' I think it starts out mysteriously. I think we should418start with a paragraph that the point of the work (really, it feels419like some serious manual labor with all the carpentry)! is to give420an equivalent formulation of RH that simply asserts that a certain421function that we will construct from counting prime powers is an422essentially square root approximation to $f(x)=x$.423424\item I wonder if we should say something right before stating RH 2425about what it means for two mathematical statements to be426equivalent? Equivalence of statements is a sort of critically427important basic tool in all of mathematical research, and is428something students encounter early on when simplifying expressions429and doing algebra. It permeates math. We touch on this also when430mention the multiplicative parity situation, where instead of giving431an equivalent statement, we give a statement that might {\em a432priori} be equivalent, but which turns out to only imply RH.433Anyway, I think there is an opportunity here.434435\item What are the frequence and amplitudes of pure C and E notes?436We could say a concrete illustrations of what we're talking about437in the section ``What do computer music files, data-compression, and prime438numbers have to do with each other?''439440\item ``But this sampling would take an enormous amount of storage441space!'' Well it would if you sampled at too many points. We might442say that to sound good it takes about xxx samples {\em per second}.443(Give the rate for audio CD's). Heh, we do say that, so rewording444this slightly might help. It might be nice to say how much space44544khz takes up, since CD's are actually uncompressed. We could say446that we're explaining why an audio CD has only about 12 sonds on it,447but exactly the same audio CD can easily hold 100 MP3's. (We say448this later...)449450\item ``Surprisingly, this seems to be roughly the way our ear451processes such a sound when we hear it.'' (in reference to storing452the spectrum, etc.) Is this a {\em biological} statement, and if so453is it the result of some research in biology that we could cite?454Otherwise, where does this assertion come from? Having us two455authors and explaining (possibly with footnotes) where all our456assertions come from I think will make our book vastly more solid457than most popular math books, which are often just full of seemingly458random unjustified statements.459460\item ``At this point we recommend to our readers that they461download...'' However, we don't recommend that they read it right462now! They should finish our book first. :-) I want our book to be a463pager turner that they can't put down. That they blow off464everything so they can finish reading it. Actually, because of465that, we should maybe put in more foreshadowing at the beginning of466this section and throughout. I want something like the paragraph at467the end of section 13 full of questions (top of page 33), but at the468beginning of section 13.469470\item (random comment) I love the idea of putting all the distracting471links in endnotes -- I'm imagining a reader that plows through our472whole book, not putting it down, not looking at footnotes, then says473``I want to read that again'', and only on a second reading of474certain parts really dives into the footnotes. Your ``Imagining475Numbers'' book was exactly like that and I think it really works.476Many popular math books are not, and it is very frustrating reading477them as a result (and they are often strongly criticized for just478this in Bulletins/Notices reviews, I think).479480\item ``So our CE chord'' -- do musicians really write ``CE'' to mean481``some combination of C and E''? I don't know. If so, we might say482``musicians write CE to mean ...'' If not, what do they write?483Should their be some notes (you know like what musicians actually484definitely do write) somewhere on our page?485486\item I think we should give lots more examples in the text like Fig48713.10 and Fig 13.11 and explain maybe something about why some of488them are valid (?).489490\item ``psycho-acoustic understanding.'' replace by a sentence saying491what that is, e.g., that humans only here certain frequencies492(etc.). Also, in that paragraph we could emphasize that a factor of49310 in compression is revolutionary -- it means you get 100 songs on494a CD instead of 12, and 200 albums on your ipod instead of 20.495496\item I wish we could end section 13 with {\em something} more, even497if it isn't at all technical. What about an illustration like Fig49818.4 (on page 45) and some sort of clever language that -- in a499nontechnical way -- explains it. It's a vivid picture. That image500shows something that looks like sound waves, and it has primes in501it. That image might be on the cover of our book. How close can we502get to it in Part 1???503504\item The Calculus Fig 15.1 -- yep, replacing it by log makes a lot of sense.505506\item We could also give a plot of a wiggly polynomial, maybe $2x^3 -5077x^2 + 5x - 2$ and its derivative $6x^2 - 14x + 5$, and note the508remarkable pattern that the derivative is got from the original509function in this case by reducing the exponents by $1$, etc. We510could remark that general observations just like this are a major511theme in calculus.512513\item Give more examples of derivatives of functions, many of which514we'll end up using later. Example derivative of constant function,515derivative of a line, derivatives of trig functions, etc.516517\item In Fig 15.2 (the graph that jumps) the axes labels are tiny.518519\item Who are this guy?: ``Newton and/or Leibniz''.520521\item ``Notice, what is happening:'' Delete the comma?522523\item Add an endnote and reference(s) for the paragraph on page 35524about distributions. What is a good reference (or references) for a525student to turn to?526527\item (**) I wonder if we should say what a ``function''. We're spending a528lot of energy saying that $\delta$ {\em isn't} a function, but we529didn't say what a function is. I didn't know an official definition530of function until my third year of undergraduate school, so the531target audience I have in mind doesn't know an ``official532definition'' either. In Calculus one typically sees sloppy things533like ``the function $1/x$ which is infinite at $0$'', so for us to534go on about delta not being a function because it is infinite is535disigeneous. Also, often us mathematicians do consider functions536$\R \to \R\cup\{\infty\}$, say. It occurs to me that the problem537with $\delta$ is not that it is not a function, but that it is not a538function that behaves well with respect to Calculus. The $\delta$539distribution is much better since e.g. $\int f \delta$ behaves so540sensibly.541542\item The caption for Figure 15.4 is totally wrong. It says ``A543picture of the derivative of a smooth graph approximating the graph544that is 1 up to some point and then 0 after that point. In each545case, the blue graph is 1 until 1 ε and 2 after 1 + .''546Wrongness: It's 4 pictures, not 1; It doesn't immediately jump from 0 to 1;547what is a ``smooth graph''? Etc. It just seems sloppy/wrong.548549\item ``Continuous approximation to the staircase $\Psi(x)$ (in red)550along with a plot (in blue) of the derivative of this [[insert 'continuous']]551approximation''552553\item (**) ``As we have hinted above, we lose no information if we554further modify our staircase by distorting the $x$-axis, replacing555$x$ by $e^t$''. We could go way slower here, and have a few556paragraphs (?) about deforming the $x$ axis by a function. We557could give several examples, pictures, etc., just like we did for558adding together two pure sounds, and I think it would help greatly559to clarify what is going on. Let's give a good specific picture and560catalogue of examples to illustrate composition functions and561thinking about what happens to their graphs. Also, in the562particular case of composing with $e^t$ isn't this just the563incredibly-familiar-in-science process of plotting data on a log564scale (or maybe exponential scale)? Every science student has565probably seen that, so it's definitely worth making that connection.566567568\item We through in a factor of $e^{t/2}$ in addition to precomposing569with $e^t$. It seems like we do the division by $e^{t/2}$ without even570commenting on that. Let's fix that. Why is it there?571572\item ``We will refer to distributions with discrete support as spike573distributions'' This language is vivid and I like it for our574booklet. Let's add an endnote though that gives the standard575terminology (e.g., ``discrete distrubtion'').576577\item ``But there are many other ways to package this vital578information, so we must explain our motivation'' If we had some even579better versions of pictures like maybe Fig 18.4 or maybe 17.4 way580earlier, we might say that explaining them is a big part of our581motivation. This is a little bit circular... but it is actually582honestly what {\em our} motivation was when we wrote this stuff up583in the first place.584585\item I'm still curiuos how people knew that Riemann computed586zeros...587588\item When we did this project initially there was a lot of excitement589as we figured out exactly who to make Figure 18.4. I think our590current text fails to convey that excitement, but I really {\em591want} to convey it. I think it's worth making our book longer and592spending more time explaining things.593594\item It is also sad that we moved $R(x)$ out of the main text. I595guess I really want part 3 back, or to make part 2 longer. I596thinked we broke it out last time due to fatigue? I'm not fatigued597anymore.598599\item ``That a simple geometric property of these zeroes...'' this600paragraph must come back into our book. It's a very nice ending, or601maybe a good motivation in the middle.602603\item ``Our aim is not even to mention complex numbers in the text,''604I'm suddenly wondering if this is at all a good constraint to put on605ourselves. I learned about -- and understood -- complex numbers in606school in 8th grade, and that was in a {\em very} small country town607in Texas. I suspect a lot of people know the basics of complex608numbers. People don't know complex {\em analysis} though. Maybe can609occassionally allow ourselves complex numbers, but definitely not610complex analysis?611612613\end{enumerate}614615616617\section{August 20, 2009}618619We have $\Phi(t) = \Psi'(e^t)/e^{t/2}$, and computed that the even and odd620Fourier transforms are:621622$${\hat \Phi}_{\rm even}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\cos(ns \log(p))$$623and the odd Fourier transform is624$${\hat \Phi}_{\rm odd}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\sin(ns \log(p)).$$625626627Plotting these to low precision (with few $p$) and one clearly "sees" the zeros of zeta influencing628the plot. Why? This seems completely mysterious. To much higher precision this disappears.629Barry suggests that instead we note that really say630${\hat \Phi}_{\rm even}(s)$ is really a distribution so it only makes sense to631integrate it against compact functions. In fact, we should have632$$633{\hat \Phi}_{\rm even}(s) = \sum_{\theta_i} \delta_{\theta_i}634$$635i.e., ${\hat \Phi}_{\rm even}(s)$ is the sum of Dirac deltas at the imaginary parts636of the zeros of the Riemann zeta function.637638Recall that $\delta_a$ is characterized by639$$640\int f(x) \delta_a dx = f(a).641$$642643So, if $f(s)$ is a function with compact support, then644$$645\int f(s) {\hat \Phi}_{\rm even}(s) = \sum_{\theta_i} f(\theta_i).646$$647Thus if we consider a function $f(s)$ with compact support in a set $S$, then648$$649\int f(s) {\hat \Phi}_{\rm even}(s) = \sum_{\theta_i\in S} f(\theta_i).650$$651E.g., suppose $S=[14,15]$. Then652$$653\int f(s) {\hat \Phi}_{\rm even}(s) = f(\theta_0).654$$655(all the integrals above are from $-\infty$ to $\infty$).656Let's say the interval $[a,b]$ contains only $\theta_0$, say.657Then658$$659\int_{\infty}^{\infty} f(s) {\hat \Phi}_{\rm even}(s)660=661\int_{a}^{b} s {\hat \Phi}_{\rm even}(s)662= \int_{a}^{b} \sum_{p^n} {\frac{\log(p)}{p^{n/2}}} s\cos(ns \log(p)) ds.663$$664It seems natural to reverse the order of integration and summation:665$$666\int_{a}^{b} \sum_{p^n} {\frac{\log(p)}{p^{n/2}}} f(s)\cos(ns \log(p)) ds667=668\sum_{p^n} {\frac{\log(p)}{p^{n/2}}} \int_{a}^{b} f(s)\cos(ns\log(p)) ds.669$$670So if we arrange that $\int_{a}^{b} f(s)\cos(s\log(p^n)) \to 0$ as671$p^n\to\infty$, we would be set. But is there any way to do this?672673674I've tried for a while, and can't find an $f(s)$ so that this converges.675676But I'm just really {\em amazed} by the mystery that677678$${\hat \Phi}_{\rm even}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\cos(ns \log(p))$$679680very visibly spikes at the zeros of zeta!681682For fun, I just tried683684$$-\sum_{p^n} {\frac{\log(p)}{p^n}}\cos(ns \log(p))$$685686and it spikes even more clearly at the zeros of zeta.687688\section{August 23, 2009}689690Things to not forget:691\begin{enumerate}692\item Thanks to Sage and everybody who has worked on it.693\item NSF grant(s)694\item Students who read drafts and gave feedback.695\end{enumerate}696697\section{August 25, 2009}698699Revisiting understanding why700701702$$-\sum_{p^n} {\frac{\log(p)}{p^n}}\cos(ns \log(p))$$703704"spikes" at the zeros of zeta.705706The obvious first thing to do is think through where this came from in the first place.707708\section{August 26, 2009}709710Notes on the current draft.711712\begin{enumerate}713\item (done) "Pure and applied" chapter title looks weird (space to left)714\item "Fourier Transforms:..." chapter title -- don't capitalize?715\item (no) Compile list of popular RH books.716\item (done) 100 figures; actually more than 125 figures717\item (done) {\tt http://wstein/rh --> http://wstein.org/rh}; also mention there that every single718figure in the entire book can be automatically computed using a Sage script available719at the above website.720\item (done, technically, but need to make nice webpage) actually post my script at http://wstein.org/rh.721\item (later) Write code to auto-translate our entire book to a web page with embedded interactive722controls, with one page for each chapter (this is more longterm).723\item (done -- but could be done better!!) We {\em have} to add the precise formulation that RH is equivalent to724$$|\pi(x) - \Li(x)| \leq \log(x) \sqrt{x}.$$725It's definitely my favorite formulation of RH.726\item "If we were to suggest some possible specific items to come home with,727after read our book, three key phrases � prime numbers, square-root accu-728rate, and error term � would head the list." You know, a fourth would be the729notion of "distribution", wouldn't it? [[replace error term by "distribution"]]730\item (done) Delete my comment "[[By prior does he mean ..." since the point as I see it731is that this is just a remark that the reader doesn't have totally understand.732\item (done) " 150 decimal digits" --> "hundreds of decimal digits" (better to be vague)733\item (done) "Offer two primes, say, P and Q each with more than 100 digits" --> few hundred digits.734I want to be vague because I don't want to get in trouble with CCR about making any specific735claims about the difficulty of cryptanalysis. Also, factoring a product of two 100 digit primes736is the sort of thing that has I think (just barely) been done by people (though it took a while and a lot of computers).737\item (done, and I added a link to Time Magazine) "In the 1990s, the Electronic" -- say that "as mentioned above, the first prize has been738claimed, but the second has not"739\item There is a standard "sketch of eratosthenes" here: http://www.teamrenzan.com/archives/writer/nagai/eratosthenes.jpg Should we include it?740\item "We become quickly stymied when we ask quite elementary questions about the741spacing of the in�nite series of prime numbers. " -- in the following paragraph we state742three questions. We should give say Richard Guy's book in an endnote as a reference, and743also mention that two of the problems have standard names.744\item "[[maybe reference in an endnote745something involving Green/Tao or Goldstein or some other recent big theorem746related to the above?]]" -- in fact, now this would be too much of a distraction. delete?747\item put the hardy-littlewood thing in as an endnote and "answer" to the challenge, though748emphasize that it is only a conjecture. the prime factorials are the most popular for awhile.749\item The plots fig\_prime\_pi\_aspect1 don't look perfect (figures 4.0.8, 4.0.9) since look slightly bent750as they are plotted using sampling. Just have to add a shade option to the function plot\_step\_function.751\item " a mere $154 = 216,\!970 - 216,\!816$; not as close as the 2004 US752elections, but pretty close nevertheless)" -- delete the election thing. Was it really off753by less than $154$? I don't think so.754\item Add endnote pointing to wikipedia PNT page, and reference Chebyshev.755\item (done) Remember to explain that we plot staircases instead of shelves, i.e., we draw a vertical riser even756though technically that is "wrong".757\item (pg 67) Add An illustration of the fund. theorem of calculus, perhaps related to the758"This is Calculus" illustration from before.759\item Cite this: The Riemann hypothesis: a resource for the afficionado and virtuoso alike760By Peter B. Borwein, Stephen Choi, Brendan Rooney761\item That $$|\pi(x) - \Li(x)| \leq \log(x) \sqrt{x}.$$ is equivalent to RH762(but with a big O) is Von Koch's theorem.763\item I think we mention Golbach in "questions about primes". In an endnote764we could mention that GRH "asymptotically" implies it. From page 84 of "The riemann hypothesis: a resource"765it says "Hardy-Littlewood prove that GRH implies almost every even integer766is a sum of two primes."767\item "My student wrote to Silva and it turns out that the record computation768of $\pi(10^{23})$ that he did took 2 months on a single computer that had7692GB of RAM."770\item IMPORTANT! I think in all the figured I really plot $-\hat{\Phi}_{\leq C}(\theta)$ -- note the771negative sign. I did that just so the spikes would be up instead of down. Soln: change label to illustrate this.772\item After reading some popular books on RH, I actually no longer want to list them in an endnote.773(e.g., Derbyshire's... )774\end{enumerate}775776777\section{August 28, 2009: Final Push}778779\begin{enumerate}780781\item (done) good distributions reference: well, the wikipedia page seems good:782\verb|http://en.wikipedia.org/wiki/Distribution_%28mathematics%29|783784\item (done) Sarnak quote in sufficing conjecture785786\item (done -- actually we did end up breaking this up) We have several natural sections already in the �rst 8 pages, but don�t break them787up as such. We should. It would make things easier to navigate. We have a section788�what are primes�. Then �prime gaps�. Then �multiplicative parity�.789790\item (done) -- [[take only first four, compress into paren remark]] Pure and applied math. I think we should double the length of791this section by adding some examples. In particular, we could use792examples all of which will appear later! Examples of possible793illustrations include:794\begin{itemize}795\item something foreshadowing Fourier theory (applied)796\item random walks (finance?) (applied)797\item data compression (applied)798\item Mersenne primes, e.g., the Lucas-Lehmer test for primality (pure mathematics)799\item Goldbach's conjecture (pure)800\end{itemize}801802\item (done) doineVERY VERY VERY for $X/2 num digits$.803804\item (done) We agree on $\pi(X)$.805806\item (done) ``These vertical dimensions might lead to a steeper ascent but807no great loss of information'' Maybe change to ``Since $\log(p)>1$,808these vertical dimensions lead to a steeper ascent but no great loss809of information.''810811812813814815\item (done) What are the frequence and amplitudes of pure C and E notes?816ANS: See http://www.phy.mtu.edu/~suits/notefreqs.html and in fact I got it right in the figures.817818\item (done -- via ref to Dave Benson's book) ``Surprisingly, this seems to be roughly the way our ear819processes such a sound when we hear it.'' (in reference to storing820the spectrum, etc.) Is this a {\em biological} statement, and if so821is it the result of some research in biology that we could cite?822Otherwise, where does this assertion come from? Having us two823authors and explaining (possibly with footnotes) where all our824assertions come from I think will make our book vastly more solid825than most popular math books, which are often just full of seemingly826random unjustified statements.827828\item (done) ``At this point we recommend to our readers that they829download...'' However, we don't recommend that they read it right830now! They should finish our book first. :-) I want our book to be a831pager turner that they can't put down. That they blow off832everything so they can finish reading it. Actually, because of833that, we should maybe put in more foreshadowing at the beginning of834this section and throughout. I want something like the paragraph at835the end of section 13 full of questions (top of page 33), but at the836beginning of section 13.837838\item (done -- you already had this) ``So our CE chord'' [["major third chord"]] -- do musicians really write ``CE'' to mean839``some combination of C and E''? I don't know. If so, we might say840``musicians write CE to mean ...'' If not, what do they write?841Should their be some notes (you know like what musicians actually842definitely do write) somewhere on our page?843844845\item (done) ``psycho-acoustic understanding.'' replace by a sentence saying846what that is, e.g., that humans only here certain frequencies847(etc.). Also, in that paragraph we could emphasize that a factor of84810 in compression is revolutionary -- it means you get 100 songs on849a CD instead of 12, and 200 albums on your ipod instead of 20.850851852\item (done) Differential Calculus, initially the creation of Newton and/or853Leibniz, acquaints us with {\em slopes} of graphs. -- have a date854855\item (done) DO THIS: This goes at the end of calculus chapter.856I wonder if we should say what a ``function''. We're spending a857lot of energy saying that $\delta$ {\em isn't} a function, but we858didn't say what a function is. I didn't know an official definition859of function until my third year of undergraduate school, so the860target audience I have in mind doesn't know an ``official861definition'' either. In Calculus one typically sees sloppy things862like ``the function $1/x$ which is infinite at $0$'', so for us to863go on about delta not being a function because it is infinite is864disigeneous. Also, often us mathematicians do consider functions865$\R \to \R\cup\{\infty\}$, say. It occurs to me that the problem866with $\delta$ is not that it is not a function, but that it is not a867function that behaves well with respect to Calculus. The $\delta$868distribution is much better since e.g. $\int f \delta$ behaves so869sensibly.870871\item (done) section to chapter872873\item Say something like "We prove nothing except one thing in text. Keep your eyes out for it. "874875\item I wonder if we should say something right before stating RH 2876about what it means for two mathematical statements to be877equivalent? -- DO this in the introduction. We give you four boxes.878We call any one of them RH. What do we mean by equivalent?879880\item Add text to two not-completely-written chapters about mathematical background not finished.881882\item Draw the sum cos but with $s'(t)$ subtracted off.883884\item - [[end note about why?]] We threw in a factor of $e^{t/2}$ in addition to precomposing885with $e^t$. It seems like we do the division by $e^{t/2}$ without even886commenting on that. Let's fix that. Why is it there?887888\end{enumerate}889890891892\end{document}893894895