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\title{RH Notebook}
\author{William Stein}
\date{}
\begin{document}
\maketitle
\tableofcontents
\section{August 18, 2009: Getting going}
I read the first 35 pages of Rockmore's book. It is like
fingernails on a chalkboard.
RH book todo:
\begin{itemize}
\item make a wiki page?
\item get my hg repo up to speed
\item read through text making list
\end{itemize}
Here is a todo list while reading the book:
\begin{enumerate}
\item Make a list of the books --both popular and not-- about
RH. (mentioned on page 1). I have Patterson (serious) and Edwards
(serious) on my desk, and Sabbagh (popular) and Rockmore
(cringe-inducing) on my desk too. There could be other popular
books that have chapters about RH that are good (or not). E.g.,
``The Millennium Problems'' by Keith Devlin has chapter 1 about RH
(the BSD chapter of that book sucks, but I maybe the RH chapter is
good?).
\item We say ``least mathematical background required'' but having
tested our booklet on students, I would say that we do not succeed
there. We could make our booklet 2 times as long and require less
math background. I've slowly come to think this would be worth it.
And we could make it much longer still by adding way more
illustrations (generated by Sage) and lots of prose explaining what
is in the illustrations (little guided tours), and this would also
be worth it.
\item I definitely want to say more in the book about how RH informs
complexity analysis in computational number theory. This is perhaps
the main way RH appears in modern computational number theory.
Maybe there are some very simple down-to-earth examples of this
principle at work.
\item I would almost like to restructure things so the illustrations
are much more extensive and integral and included in the main text.
Then the additional Sage interacts are merely an ``additional
resource'' for those wishing to investigate further. They're an
added bonus. But they can also be safely ignored.
\item Typo: ``websitte''.
\item I think we should remove the business about how long it should
take to read the book. Let the reader start reading and decide for
themselves. Otherwise, they might feel insecure and wonder all the
time if they are taking ``too long''.
\item ``less than 100, 10,000, 1,000,000, `` that looks at first
glance like a single huge number. Maybe make it three statements.
less than 100? less than 10,000? less than 1,000,000?
\item Picture of Bott?
\item Picture of Zagier?
\item This sentence: ``If we are to believe Aristotle, the early
Pythagoreans thought that the principles governing Number are “the
principles of all things,” the elements of number being more basic
than the Empedoclean physical elements earth, air, fire, water.''
I've been looking at other popular math books, and they never just
assume the reader knows who Aristotle is, Pythagoreans were, or what
Empedoclean means. In fact, I have no idea what Empedoclean means,
and I can easily forgot {\em when} the Pythagoreans were around.
That said, I would rather say nothing to say something wrong.
\item Descarte picture? Are there any? [[Yes -- see Wikipedia]]
\item Speaking of ``wrong'' (see above), somebody emailed me this:
\begin{verbatim}
In a text "Elementary Number Theory" in section 7.1.2, you have
an implementation of the sieve of Eratosthense. Melissa O'Neill
wrote a paper, "The Genuine Sieve of Eratosthenese". I do not
believe that your program meets here criteria for being the
genuine sieve of Eratosthenses. I used IDLE on an IBM/PC to
run your program and crashed, if I entered the value of 200000.
I can create a list of primes at least up to 200000 if I use a
Python program that meets her criteria.
\end{verbatim}
We should read Melissa O'Neill's paper to see what the deal is.
\item ``Contemporary physicists dream of a “final theory.”'' Do they
really? In what sense?
\item ``Don Quixote encountered this...'': Who is he exactly? A
fictional character, a person? When? I've heard of him, but
honestly I've never read anything nontrivial about him, and I doubt
most of our readers will have. They might see him mainly as a
mysterious person whose last name is hard to pronounce.
\item Why exactly do Cicada's come out every 17 years? I saw Bruce
Jordan in Princeton recently and we started talking about this
(since they have Cicada's there), and I quickly realized I didn't
really have a clue.
\item ``Philolaus (a predecessor of Plato)'' that isn't a good enough
introduction to Philolaus, given that it is the first mention of
Plato. Again, many readers might not know Plato so well. Heck, I
don't. I view all the above remarks as opportunities to expand our
book's readership and mission a bit, rather than criticisms of it.
\item ``But, until Euclid, prime numbers seem not to have been singled
out as the extraordinary math- ematical concept, central to any deep
understanding of numerical phenomena, that they are now understood
to be.'' Here we are foreshadowing Euclid's proof that there are
infinitely many primes, etc. But this is also the first time Euclid
is even mentioned. To a casual reader it just feels that it's a
point in an outline that hasn't been filled in.
\item Instead of starting with the 300 factoring example, perhaps we
should first start with a smaller one where we can list {\em every}
single factorization tree/order. This makes things feel less {\em
abstract}, since the reader doesn't have to imagine all the
missing factorizations.
\item ``more than 100 digits, to your computing machine and ask it to
multiply them together: you will get their product N = P × Q with
its 200 or so digits in a few microseconds.'' I just checked and it
is a few hundred {\em nanoseconds} to do that. So lets change to
``few hundred nanoseconds'' or perhaps better ``a just under a
microsecond.''
\item Our proof of the infinitude of primes on page 8 is the first
time in the book we use symbolic notation, give a proof, reason
abstractly, etc. I wonder if we could do a little more to prepare
the reader. I just read Rockmore's horrendous proof of the same
thing in his book on RH -- it's pages of tedium to say in words what
takes 1 second with symbols. But I'm attracted to the challenge of
doing something a little bit in between, e.g., having an example.
\item Move our discussion of EFF cash prize up, since the prize was
just won! I wonder if there are any press releases about the prize
being awareded, which we could cite or point to?
\item Here is Sage actually computing the decimal digits of the
biggest known Mersenne prime:
\begin{verbatim}
sage: time a =2^43112609-1
CPU times: user 0.01 s, sys: 0.01 s, total: 0.01 s
Wall time: 0.02 s
sage: time s = str(a)
CPU times: user 12.23 s, sys: 0.99 s, total: 13.22 s
Wall time: 13.63 s
sage: s[-10:]
'6697152511'
sage: time sum(a.digits())
CPU times: user 15.25 s, sys: 1.07 s, total: 16.33 s
Wall time: 16.84 s
58416637
\end{verbatim}
\item ``But there is no obvious way'' -- maybe ``no known way''?
\item ``In Figure 3.3 we use the primes 2, 3, 5, and 7 to sieve out
the primes up to 100, where instead of crossing out multiples we
grey them out, and instead of circling primes we color their box
red.'' I could make a sequence of figures where we do cross them
out too? The grey background is hard to see and probably hard to
print, so I can do better there too.
\item For all these questions: ``Are there infinitely many pairs of
primes whose difference is 4? Answer: equally unknown. Is every even
number greater than 2 a sum of two primes? Answer: unknown. Are
there infinitely many primes which are 1 more than a perfect square?
Answer: unknown.'' we could give precise references into Richard
Guy's book ``Unsolved Problems in Number theory'', which in turn has
a very good collection of references and more detailed description
of each problem. This would be a good endnote.
\item I may as well draw a plot of $\text{Gap}_k(X)$ for various k together
on one plot.
\item We have several natural {\em sections} already in the first 8
pages, but don't break them up as such. We should. It would make
things easier to navigate. We have a section ``what are primes''. Then ``prime gaps''. Then ``multiplicative parity''.
\item ``Here is some data:''... and a weird big page break?
\item On page 12, the references to Borwein etc. should of course be
moved to an endnote.
\item On page 15 (Fig 5.4), it would be nice to have a less zoomed out
big figure for starters, to look at while reading along. Basically
like Fig 5.5, which looks very nice. Those figures could be bigger
too. I really like this part of the text though, where we are
spending a lot of care explaining the mathematics.
\item I need to figure out how to be very precise in placing all the
figures where we want, not where latex wants. Right now there
placement significantly detracts from readability.
\item ``The particular issue before us is, in our opinion, twofold,
both applied, and pure: can we curve-fit the “staircase of primes”
by a well approximating smooth curve?'' I think it would be worth
emphasizing that our smooth curve must be given for a ``formula''.
I mean, a typical reader might just think ``of course any kid could
take a pencil and draw a smooth curve through the staircase of
primes''. But to get a curve given by an sort of analytic formula
at all and which happens to have {\em anything } at all to do with
the function $\pi(X)$ -- well that seems really hard. A typical
reader might have no idea where to start to do that. Maybe we can
express this somehow?
\item `` the chances that a number N is a prime is inversely
proportional to the number of digits of N''. Does there exist any
heuristic plausibility argument for this assumption that would make
sense to give at this point? Things are made a bit confusing since
the constant isn't 1, e.g., the probability that a number around a
billion is prime is not about ``1 in 9''.
\item Pure and applied math. I think we should double the length of
this section by adding some examples. In particular, we could use
examples all of which will appear later! Examples of possible
illustrations include:
\begin{itemize}
\item something foreshadowing Fourier theory (applied)
\item random walks (finance?) (applied)
\item data compression (applied)
\item Mersenne primes, e.g., the Lucas-Lehmer test for primality (pure mathematics)
\item Goldbach's conjecture (pure)
\item The Hardy-Littlewood conjecture about asymptotics of $\text{Gap}_k(X)$. (pure)
\item Complex numbers (pure and applied); and with an endnote that
points at your book Imagining Numbers?
\end{itemize}
\item Picture of Gauss (and I really like our Gauss dates)
\item ``Roughly speaking, this means that the number of primes up to X
is X times the reciprocal of 2.3 times the number of digits of X .''
I think this is confusing to read. The reciprocal of 2.3 is kind of
funny, since 2.3 is already mysterious. It's really $1/log_e(10)$,
which is $0.43429448190325176...$, or basically $.4$. Maybe better
would be ``Very roughly speaking, this means that the number of
primes up to $X$ is about $X$ divided by twice the number of digits
of $X$.'' We can make a
table to illustrate this further, but also to emphasize that it's
not that close. Something like this:
\begin{verbatim}
sage: for i in [2..10]: print i, prime_pi(10^i), floor((10^i-1)/(i*2))
....:
2 25 24
3 168 166
4 1229 1249
5 9592 9999
6 78498 83333
7 664579 714285
8 5761455 6249999
9 50847534 55555555
10 455052511 499999999
\end{verbatim}
It's kind of convenient that for $99$, $999$, and $9999$, the
approximations got by taking ``$X$ divided by twice the number of
digits of $X$ are very close to $\pi(X)$. Anyway, rounding to $0.5$
instead of $0.43$ makes it really simple to describe.
\item We might do something to warn our reader that if they see
``$\log(X)$'' they shouldn't run in fear and think ``holy crud, I
have no idea what log is and I never understand that in high
school,'' since we are about to explain it. I'm imagining say my
brother as reading this -- he literally probably hasn't seen log
once in a decade though he is good with numbers (running five
businesses in San Diego). He told me that when he sees a page with
a mathematical formula involving symbols he doesn't know, he'll just
block it out. So if we sneak ``log'' in to a sentence or two before
we use it in a formula, it'll get by that filter (which is probably
pretty common with non-math people).
I'm imagining a solution like this:
\begin{itemize}
\item We figure out where logs first came from and give one sentence about
this (I think they arose in doing arithmetic efficiently?)
\item We demystify log {\em before} using it in any formula by explaining
that it is ``about twice the number of digits''.
\end{itemize}
One other issue is that in much of math education, unfortunately $\ln$
means natural log and $\log$ means $\log$-base-10. It's really
annoying... We can mention this somehow.
It might be worthwhile to remark that $e^x$ is the unique nonzero
function that equals its own derivative -- perhaps this is a way to
sneak in a mention of derivatives before later in the book where we
use them a lot more. Anyway there are two issues: (1) what are logs,
and (2) what is this ``natural log''?
\item ``the 2004 US elections'' -- this will not be in people's minds
for a {\em book} so much. It may be better to remove or expand with
a statement about just how close they were with a reference. E.g.,
``the 2004 US elections, in which ... beat ... by a mere
... votes!'' Wasn't the 2000 election even closer, or am I
mis-remembering?
\item ``So when Gauss thought his curve missed...'' let's compute the
square root explicitly here, i.e., just spell this out some more
(instead of leaving an exercise for the reader).
\item ``devil fable'' I found this graphic via an image search on google:
\url{http://blog.al.com/stantis/2007/11/Stantis-Devil
If 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...
\item We write ``$\pi(X)$ for various large numbers $N$'' in our devil fable. Oops.
\item We should draw an illustration of the checklist in the story.
It would be easy.
\item We make the claim ``The average error (over-counted or
undercounted) would be proportional to $\sqrt{N}$.'' We do not
justify this claim at all. We might say that it follows from a
result about random walks. (Does it really follow from the central
limit theorem somehow?) Also, given that we assumed that the error
rate is 0.001\% can't we say what the constant in the proportion is?
Also, I think we could give an estimate of how far they would be off
for $N=3,000,000$. We could deduce Gauss's error rate, right?
\item In figure 10.1 with plots of Li, pi, and X/Log(X), I should
put labels in the actual plot. It is lazy putting them only
in the caption.
\item I should update the $X=4\cdot 10^{22}$ to whatever
the current record is, I think $10^{24}$, maybe. And also update
the reference, which may be wrong. Also, here is where we can possibly
discuss how to compute $\pi(X)$, or if not we can at least point
to an (extended) endnote. When this is done, be sure to search and
update all other references to $4\cdot 10^{22}$.
\item We write -- `` an easier fact, which follows directly from
elementary calculus'' for the fact that $\Li(x)$ is asymptotic to
$X/\log(X)$. We should prove this rigorously in an endnote.
\item ``It was proved in 1896 indepdently by Hadamard and de la Valle Poussin. ``
(1) typo in ``indepdently''; we should say something about who these guys are,
and give links to Wikipedia (say).
\item We write ``is much deeper than the Prime Number Theorem''. I
think the phrase ``is much deeper'' is mathematical jargon, because
popular math books would often have a little interlude to say
something about what deep means to mathematicians. It's basically
``difficult and any proof will use and influence a wide range of
mathematics''. So we too can add a little more to emphasize what we
mean by the word ``deep''. Or we can just say that the rest of this
paragraph explains what we mean (indeed, it does). Maybe everything
is perfect as is.
\item ``It is the kind of conjecture that Frans Oort...'' let's have a
sentence about who Oort is. E.g., Dutch mathematician, born 19xx,
student of xxx... I might have a picture of him too.
\item We write ``A proof of RH would, therefore, fall into the applied
category, given our discussion above.'' But we changed our
discussion above, so this is no longer quite true.
\item I wonder if I could draw a 3d picture of an actual staircase
whose side profile is the plot of $\pi(X)$, but rendered at an angle
to look like a real staircase. This might be a nice illustration.
\item In the section ``Tinkering with the carpentry of the staircase
of primes.'' I should draw several plots illustrating every single
one of the steps we discuss about tinkering with the staircase.
\item ``These vertical dimensions might lead to a steeper ascent but
no great loss of information'' Maybe change to ``Since $\log(p)>1$,
these vertical dimensions lead to a steeper ascent but no great loss
of information.''
\item ``Do not worry if you do not understand why our first and second
formulations of Riemann's Hypothesis are equivalent.'' We should
either rigorously prove this in an endnote (my preference at the
moment) or gave a reference that totally does it. I could imagine a
better student who has a more advanced background, who would benefit
by seeing a proof at this point. And it might help us keep things
straight... e.g., we had this equivalence wrong I think in some
version of our notes long ago.
\item ``variety of equivalent ways we have to express Riemann’s
propose answers to the question'' -- I think ``propose'' should be
``proposed''.
\item I'm worried that our second statement of RH is possibly
confusing because it says ``This new staircase is essentially square
root close''. However, given a line and curve the notion of close
is vague. What we really mean is that the function $\psi$ given by
the new staircase is an essentially square root approximation to the
function $f(x) = x$.
\item Having just read ``Tinkering with the carpentry of the staircase
of primes.'' I think it starts out mysteriously. I think we should
start with a paragraph that the point of the work (really, it feels
like some serious manual labor with all the carpentry)! is to give
an equivalent formulation of RH that simply asserts that a certain
function that we will construct from counting prime powers is an
essentially square root approximation to $f(x)=x$.
\item I wonder if we should say something right before stating RH 2
about what it means for two mathematical statements to be
equivalent? Equivalence of statements is a sort of critically
important basic tool in all of mathematical research, and is
something students encounter early on when simplifying expressions
and doing algebra. It permeates math. We touch on this also when
mention the multiplicative parity situation, where instead of giving
an equivalent statement, we give a statement that might {\em a
priori} be equivalent, but which turns out to only imply RH.
Anyway, I think there is an opportunity here.
\item What are the frequence and amplitudes of pure C and E notes?
We could say a concrete illustrations of what we're talking about
in the section ``What do computer music files, data-compression, and prime
numbers have to do with each other?''
\item ``But this sampling would take an enormous amount of storage
space!'' Well it would if you sampled at too many points. We might
say that to sound good it takes about xxx samples {\em per second}.
(Give the rate for audio CD's). Heh, we do say that, so rewording
this slightly might help. It might be nice to say how much space
44khz takes up, since CD's are actually uncompressed. We could say
that we're explaining why an audio CD has only about 12 sonds on it,
but exactly the same audio CD can easily hold 100 MP3's. (We say
this later...)
\item ``Surprisingly, this seems to be roughly the way our ear
processes such a sound when we hear it.'' (in reference to storing
the spectrum, etc.) Is this a {\em biological} statement, and if so
is it the result of some research in biology that we could cite?
Otherwise, where does this assertion come from? Having us two
authors and explaining (possibly with footnotes) where all our
assertions come from I think will make our book vastly more solid
than most popular math books, which are often just full of seemingly
random unjustified statements.
\item ``At this point we recommend to our readers that they
download...'' However, we don't recommend that they read it right
now! They should finish our book first. :-) I want our book to be a
pager turner that they can't put down. That they blow off
everything so they can finish reading it. Actually, because of
that, we should maybe put in more foreshadowing at the beginning of
this section and throughout. I want something like the paragraph at
the end of section 13 full of questions (top of page 33), but at the
beginning of section 13.
\item (random comment) I love the idea of putting all the distracting
links in endnotes -- I'm imagining a reader that plows through our
whole book, not putting it down, not looking at footnotes, then says
``I want to read that again'', and only on a second reading of
certain parts really dives into the footnotes. Your ``Imagining
Numbers'' book was exactly like that and I think it really works.
Many popular math books are not, and it is very frustrating reading
them as a result (and they are often strongly criticized for just
this in Bulletins/Notices reviews, I think).
\item ``So our CE chord'' -- do musicians really write ``CE'' to mean
``some combination of C and E''? I don't know. If so, we might say
``musicians write CE to mean ...'' If not, what do they write?
Should their be some notes (you know like what musicians actually
definitely do write) somewhere on our page?
\item I think we should give lots more examples in the text like Fig
13.10 and Fig 13.11 and explain maybe something about why some of
them are valid (?).
\item ``psycho-acoustic understanding.'' replace by a sentence saying
what that is, e.g., that humans only here certain frequencies
(etc.). Also, in that paragraph we could emphasize that a factor of
10 in compression is revolutionary -- it means you get 100 songs on
a CD instead of 12, and 200 albums on your ipod instead of 20.
\item I wish we could end section 13 with {\em something} more, even
if it isn't at all technical. What about an illustration like Fig
18.4 (on page 45) and some sort of clever language that -- in a
nontechnical way -- explains it. It's a vivid picture. That image
shows something that looks like sound waves, and it has primes in
it. That image might be on the cover of our book. How close can we
get to it in Part 1???
\item The Calculus Fig 15.1 -- yep, replacing it by log makes a lot of sense.
\item We could also give a plot of a wiggly polynomial, maybe $2x^3 -
7x^2 + 5x - 2$ and its derivative $6x^2 - 14x + 5$, and note the
remarkable pattern that the derivative is got from the original
function in this case by reducing the exponents by $1$, etc. We
could remark that general observations just like this are a major
theme in calculus.
\item Give more examples of derivatives of functions, many of which
we'll end up using later. Example derivative of constant function,
derivative of a line, derivatives of trig functions, etc.
\item In Fig 15.2 (the graph that jumps) the axes labels are tiny.
\item Who are this guy?: ``Newton and/or Leibniz''.
\item ``Notice, what is happening:'' Delete the comma?
\item Add an endnote and reference(s) for the paragraph on page 35
about distributions. What is a good reference (or references) for a
student to turn to?
\item (**) I wonder if we should say what a ``function''. We're spending a
lot of energy saying that $\delta$ {\em isn't} a function, but we
didn't say what a function is. I didn't know an official definition
of function until my third year of undergraduate school, so the
target audience I have in mind doesn't know an ``official
definition'' either. In Calculus one typically sees sloppy things
like ``the function $1/x$ which is infinite at $0$'', so for us to
go on about delta not being a function because it is infinite is
disigeneous. Also, often us mathematicians do consider functions
$\R \to \R\cup\{\infty\}$, say. It occurs to me that the problem
with $\delta$ is not that it is not a function, but that it is not a
function that behaves well with respect to Calculus. The $\delta$
distribution is much better since e.g. $\int f \delta$ behaves so
sensibly.
\item The caption for Figure 15.4 is totally wrong. It says ``A
picture of the derivative of a smooth graph approximating the graph
that is 1 up to some point and then 0 after that point. In each
case, the blue graph is 1 until 1 ε and 2 after 1 + .''
Wrongness: It's 4 pictures, not 1; It doesn't immediately jump from 0 to 1;
what is a ``smooth graph''? Etc. It just seems sloppy/wrong.
\item ``Continuous approximation to the staircase $\Psi(x)$ (in red)
along with a plot (in blue) of the derivative of this [[insert 'continuous']]
approximation''
\item (**) ``As we have hinted above, we lose no information if we
further modify our staircase by distorting the $x$-axis, replacing
$x$ by $e^t$''. We could go way slower here, and have a few
paragraphs (?) about deforming the $x$ axis by a function. We
could give several examples, pictures, etc., just like we did for
adding together two pure sounds, and I think it would help greatly
to clarify what is going on. Let's give a good specific picture and
catalogue of examples to illustrate composition functions and
thinking about what happens to their graphs. Also, in the
particular case of composing with $e^t$ isn't this just the
incredibly-familiar-in-science process of plotting data on a log
scale (or maybe exponential scale)? Every science student has
probably seen that, so it's definitely worth making that connection.
\item We through in a factor of $e^{t/2}$ in addition to precomposing
with $e^t$. It seems like we do the division by $e^{t/2}$ without even
commenting on that. Let's fix that. Why is it there?
\item ``We will refer to distributions with discrete support as spike
distributions'' This language is vivid and I like it for our
booklet. Let's add an endnote though that gives the standard
terminology (e.g., ``discrete distrubtion'').
\item ``But there are many other ways to package this vital
information, so we must explain our motivation'' If we had some even
better versions of pictures like maybe Fig 18.4 or maybe 17.4 way
earlier, we might say that explaining them is a big part of our
motivation. This is a little bit circular... but it is actually
honestly what {\em our} motivation was when we wrote this stuff up
in the first place.
\item I'm still curiuos how people knew that Riemann computed
zeros...
\item When we did this project initially there was a lot of excitement
as we figured out exactly who to make Figure 18.4. I think our
current text fails to convey that excitement, but I really {\em
want} to convey it. I think it's worth making our book longer and
spending more time explaining things.
\item It is also sad that we moved $R(x)$ out of the main text. I
guess I really want part 3 back, or to make part 2 longer. I
thinked we broke it out last time due to fatigue? I'm not fatigued
anymore.
\item ``That a simple geometric property of these zeroes...'' this
paragraph must come back into our book. It's a very nice ending, or
maybe a good motivation in the middle.
\item ``Our aim is not even to mention complex numbers in the text,''
I'm suddenly wondering if this is at all a good constraint to put on
ourselves. I learned about -- and understood -- complex numbers in
school in 8th grade, and that was in a {\em very} small country town
in Texas. I suspect a lot of people know the basics of complex
numbers. People don't know complex {\em analysis} though. Maybe can
occassionally allow ourselves complex numbers, but definitely not
complex analysis?
\end{enumerate}
\section{August 20, 2009}
We have $\Phi(t) = \Psi'(e^t)/e^{t/2}$, and computed that the even and odd
Fourier transforms are:
$${\hat \Phi}_{\rm even}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\cos(ns \log(p))$$
and the odd Fourier transform is
$${\hat \Phi}_{\rm odd}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\sin(ns \log(p)).$$
Plotting these to low precision (with few $p$) and one clearly "sees" the zeros of zeta influencing
the plot. Why? This seems completely mysterious. To much higher precision this disappears.
Barry suggests that instead we note that really say
${\hat \Phi}_{\rm even}(s)$ is really a distribution so it only makes sense to
integrate it against compact functions. In fact, we should have
$$
{\hat \Phi}_{\rm even}(s) = \sum_{\theta_i} \delta_{\theta_i}
$$
i.e., ${\hat \Phi}_{\rm even}(s)$ is the sum of Dirac deltas at the imaginary parts
of the zeros of the Riemann zeta function.
Recall that $\delta_a$ is characterized by
$$
\int f(x) \delta_a dx = f(a).
$$
So, if $f(s)$ is a function with compact support, then
$$
\int f(s) {\hat \Phi}_{\rm even}(s) = \sum_{\theta_i} f(\theta_i).
$$
Thus if we consider a function $f(s)$ with compact support in a set $S$, then
$$
\int f(s) {\hat \Phi}_{\rm even}(s) = \sum_{\theta_i\in S} f(\theta_i).
$$
E.g., suppose $S=[14,15]$. Then
$$
\int f(s) {\hat \Phi}_{\rm even}(s) = f(\theta_0).
$$
(all the integrals above are from $-\infty$ to $\infty$).
Let's say the interval $[a,b]$ contains only $\theta_0$, say.
Then
$$
\int_{\infty}^{\infty} f(s) {\hat \Phi}_{\rm even}(s)
=
\int_{a}^{b} s {\hat \Phi}_{\rm even}(s)
= \int_{a}^{b} \sum_{p^n} {\frac{\log(p)}{p^{n/2}}} s\cos(ns \log(p)) ds.
$$
It seems natural to reverse the order of integration and summation:
$$
\int_{a}^{b} \sum_{p^n} {\frac{\log(p)}{p^{n/2}}} f(s)\cos(ns \log(p)) ds
=
\sum_{p^n} {\frac{\log(p)}{p^{n/2}}} \int_{a}^{b} f(s)\cos(ns\log(p)) ds.
$$
So if we arrange that $\int_{a}^{b} f(s)\cos(s\log(p^n)) \to 0$ as
$p^n\to\infty$, we would be set. But is there any way to do this?
I've tried for a while, and can't find an $f(s)$ so that this converges.
But I'm just really {\em amazed} by the mystery that
$${\hat \Phi}_{\rm even}(s): = \sum_{p^n} {\frac{\log(p)}{p^{n/2}}}\cos(ns \log(p))$$
very visibly spikes at the zeros of zeta!
For fun, I just tried
$$-\sum_{p^n} {\frac{\log(p)}{p^n}}\cos(ns \log(p))$$
and it spikes even more clearly at the zeros of zeta.
\section{August 23, 2009}
Things to not forget:
\begin{enumerate}
\item Thanks to Sage and everybody who has worked on it.
\item NSF grant(s)
\item Students who read drafts and gave feedback.
\end{enumerate}
\section{August 25, 2009}
Revisiting understanding why
$$-\sum_{p^n} {\frac{\log(p)}{p^n}}\cos(ns \log(p))$$
"spikes" at the zeros of zeta.
The obvious first thing to do is think through where this came from in the first place.
\section{August 26, 2009}
Notes on the current draft.
\begin{enumerate}
\item (done) "Pure and applied" chapter title looks weird (space to left)
\item "Fourier Transforms:..." chapter title -- don't capitalize?
\item (no) Compile list of popular RH books.
\item (done) 100 figures; actually more than 125 figures
\item (done) {\tt http://wstein/rh --> http://wstein.org/rh}; also mention there that every single
figure in the entire book can be automatically computed using a Sage script available
at the above website.
\item (done, technically, but need to make nice webpage) actually post my script at http://wstein.org/rh.
\item (later) Write code to auto-translate our entire book to a web page with embedded interactive
controls, with one page for each chapter (this is more longterm).
\item (done -- but could be done better!!) We {\em have} to add the precise formulation that RH is equivalent to
$$|\pi(x) - \Li(x)| \leq \log(x) \sqrt{x}.$$
It's definitely my favorite formulation of RH.
\item "If we were to suggest some possible specific items to come home with,
after read our book, three key phrases � prime numbers, square-root accu-
rate, and error term � would head the list." You know, a fourth would be the
notion of "distribution", wouldn't it? [[replace error term by "distribution"]]
\item (done) Delete my comment "[[By prior does he mean ..." since the point as I see it
is that this is just a remark that the reader doesn't have totally understand.
\item (done) " 150 decimal digits" --> "hundreds of decimal digits" (better to be vague)
\item (done) "Offer two primes, say, P and Q each with more than 100 digits" --> few hundred digits.
I want to be vague because I don't want to get in trouble with CCR about making any specific
claims about the difficulty of cryptanalysis. Also, factoring a product of two 100 digit primes
is the sort of thing that has I think (just barely) been done by people (though it took a while and a lot of computers).
\item (done, and I added a link to Time Magazine) "In the 1990s, the Electronic" -- say that "as mentioned above, the first prize has been
claimed, but the second has not"
\item There is a standard "sketch of eratosthenes" here: http://www.teamrenzan.com/archives/writer/nagai/eratosthenes.jpg Should we include it?
\item "We become quickly stymied when we ask quite elementary questions about the
spacing of the in�nite series of prime numbers. " -- in the following paragraph we state
three questions. We should give say Richard Guy's book in an endnote as a reference, and
also mention that two of the problems have standard names.
\item "[[maybe reference in an endnote
something involving Green/Tao or Goldstein or some other recent big theorem
related to the above?]]" -- in fact, now this would be too much of a distraction. delete?
\item put the hardy-littlewood thing in as an endnote and "answer" to the challenge, though
emphasize that it is only a conjecture. the prime factorials are the most popular for awhile.
\item The plots fig\_prime\_pi\_aspect1 don't look perfect (figures 4.0.8, 4.0.9) since look slightly bent
as they are plotted using sampling. Just have to add a shade option to the function plot\_step\_function.
\item " a mere $154 = 216,\!970 - 216,\!816$; not as close as the 2004 US
elections, but pretty close nevertheless)" -- delete the election thing. Was it really off
by less than $154$? I don't think so.
\item Add endnote pointing to wikipedia PNT page, and reference Chebyshev.
\item (done) Remember to explain that we plot staircases instead of shelves, i.e., we draw a vertical riser even
though technically that is "wrong".
\item (pg 67) Add An illustration of the fund. theorem of calculus, perhaps related to the
"This is Calculus" illustration from before.
\item Cite this: The Riemann hypothesis: a resource for the afficionado and virtuoso alike
By Peter B. Borwein, Stephen Choi, Brendan Rooney
\item That $$|\pi(x) - \Li(x)| \leq \log(x) \sqrt{x}.$$ is equivalent to RH
(but with a big O) is Von Koch's theorem.
\item I think we mention Golbach in "questions about primes". In an endnote
we could mention that GRH "asymptotically" implies it. From page 84 of "The riemann hypothesis: a resource"
it says "Hardy-Littlewood prove that GRH implies almost every even integer
is a sum of two primes."
\item "My student wrote to Silva and it turns out that the record computation
of $\pi(10^{23})$ that he did took 2 months on a single computer that had
2GB of RAM."
\item IMPORTANT! I think in all the figured I really plot $-\hat{\Phi}_{\leq C}(\theta)$ -- note the
negative sign. I did that just so the spikes would be up instead of down. Soln: change label to illustrate this.
\item After reading some popular books on RH, I actually no longer want to list them in an endnote.
(e.g., Derbyshire's... )
\end{enumerate}
\section{August 28, 2009: Final Push}
\begin{enumerate}
\item (done) good distributions reference: well, the wikipedia page seems good:
\verb|http://en.wikipedia.org/wiki/Distribution_
\item (done) Sarnak quote in sufficing conjecture
\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 them
up as such. We should. It would make things easier to navigate. We have a section
�what are primes�. Then �prime gaps�. Then �multiplicative parity�.
\item (done) -- [[take only first four, compress into paren remark]] Pure and applied math. I think we should double the length of
this section by adding some examples. In particular, we could use
examples all of which will appear later! Examples of possible
illustrations include:
\begin{itemize}
\item something foreshadowing Fourier theory (applied)
\item random walks (finance?) (applied)
\item data compression (applied)
\item Mersenne primes, e.g., the Lucas-Lehmer test for primality (pure mathematics)
\item Goldbach's conjecture (pure)
\end{itemize}
\item (done) doineVERY VERY VERY for $X/2 num digits$.
\item (done) We agree on $\pi(X)$.
\item (done) ``These vertical dimensions might lead to a steeper ascent but
no great loss of information'' Maybe change to ``Since $\log(p)>1$,
these vertical dimensions lead to a steeper ascent but no great loss
of information.''
\item (done) What are the frequence and amplitudes of pure C and E notes?
ANS: See http://www.phy.mtu.edu/~suits/notefreqs.html and in fact I got it right in the figures.
\item (done -- via ref to Dave Benson's book) ``Surprisingly, this seems to be roughly the way our ear
processes such a sound when we hear it.'' (in reference to storing
the spectrum, etc.) Is this a {\em biological} statement, and if so
is it the result of some research in biology that we could cite?
Otherwise, where does this assertion come from? Having us two
authors and explaining (possibly with footnotes) where all our
assertions come from I think will make our book vastly more solid
than most popular math books, which are often just full of seemingly
random unjustified statements.
\item (done) ``At this point we recommend to our readers that they
download...'' However, we don't recommend that they read it right
now! They should finish our book first. :-) I want our book to be a
pager turner that they can't put down. That they blow off
everything so they can finish reading it. Actually, because of
that, we should maybe put in more foreshadowing at the beginning of
this section and throughout. I want something like the paragraph at
the end of section 13 full of questions (top of page 33), but at the
beginning of section 13.
\item (done -- you already had this) ``So our CE chord'' [["major third chord"]] -- do musicians really write ``CE'' to mean
``some combination of C and E''? I don't know. If so, we might say
``musicians write CE to mean ...'' If not, what do they write?
Should their be some notes (you know like what musicians actually
definitely do write) somewhere on our page?
\item (done) ``psycho-acoustic understanding.'' replace by a sentence saying
what that is, e.g., that humans only here certain frequencies
(etc.). Also, in that paragraph we could emphasize that a factor of
10 in compression is revolutionary -- it means you get 100 songs on
a CD instead of 12, and 200 albums on your ipod instead of 20.
\item (done) Differential Calculus, initially the creation of Newton and/or
Leibniz, acquaints us with {\em slopes} of graphs. -- have a date
\item (done) DO THIS: This goes at the end of calculus chapter.
I wonder if we should say what a ``function''. We're spending a
lot of energy saying that $\delta$ {\em isn't} a function, but we
didn't say what a function is. I didn't know an official definition
of function until my third year of undergraduate school, so the
target audience I have in mind doesn't know an ``official
definition'' either. In Calculus one typically sees sloppy things
like ``the function $1/x$ which is infinite at $0$'', so for us to
go on about delta not being a function because it is infinite is
disigeneous. Also, often us mathematicians do consider functions
$\R \to \R\cup\{\infty\}$, say. It occurs to me that the problem
with $\delta$ is not that it is not a function, but that it is not a
function that behaves well with respect to Calculus. The $\delta$
distribution is much better since e.g. $\int f \delta$ behaves so
sensibly.
\item (done) section to chapter
\item Say something like "We prove nothing except one thing in text. Keep your eyes out for it. "
\item I wonder if we should say something right before stating RH 2
about what it means for two mathematical statements to be
equivalent? -- DO this in the introduction. We give you four boxes.
We call any one of them RH. What do we mean by equivalent?
\item Add text to two not-completely-written chapters about mathematical background not finished.
\item Draw the sum cos but with $s'(t)$ subtracted off.
\item - [[end note about why?]] We threw in a factor of $e^{t/2}$ in addition to precomposing
with $e^t$. It seems like we do the division by $e^{t/2}$ without even
commenting on that. Let's fix that. Why is it there?
\end{enumerate}
\end{document}
