Math 640: EXPERIMENTAL MATHEMATICS
Spring 2018 (Rutgers University) Webpage
http://sites.math.rutgers.edu/~zeilberg/math640_18.html
Last Update: April 30, 2018.
- Teacher:
Dr. Doron ZEILBERGER ("Dr. Z")
-
Classroom:
Allison Road Classroom Building
[Busch Campus], IML Room 116.
[Except Jan. 25, 2018, where it is at Hill 425, and we would have guest-lecturer Dr. Neil Sloane ]
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Time: Mondays and Thursdays , period 3 (12:00noon-1:20pm)
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Main Textbook: "Lectures on the Riemann Zeta Function"
by Henryk Iwaniec.
Other texts used:
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Donald Newman's short proof by Don Zagier
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Bernhard Riemman's memoir (English translation)
[original,
original mansuscript]
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proof of the modularity of the theta function, scanned
from pp. 52-53 of the classic book "Fourier Series and Integrals" by Dr. Z. academic father and grandfather (Harry Dym and Henry McKean respectively).
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George Polya's beautiful elementary proof, scanned
from pp. 40-41 of the classic book "A brief introduction to Theta Functions" by Richard Bellman.
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The first 100000 zeros of Zeta(s), (downloaded from the Andrew Odlyzko's website)
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The first 100000 zeros of Zeta(s), (as a Maple list called Z10) [Thanks for
Ahshan Kahn for the conversion).
- Brendan W. Sullivan's fascinating article on
"Numerous Proofs that ζ(2)=π2/2"
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Dr. Z's gorgeous combinatorial proof of Newton's identitities.
Recall that Newton's identities, generalized to "infinite" polynomials, enabled Euler to evaluate Zeta(2n).
[For a determinant formula for the power sums, pn in terrms of the elementary symmetric functions, en, see p.28 of
Ian Macdonald's classic book, "Symmmetric Functions".]
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Chris Houghes' nice article on the moments of the Riemann Zeta function and Random Matrix Theory
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The wikipedia article on Hough Montgomery's Pair Correlation Conjecture.
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Atle Selberg's seminal paper on the zeros of Zeta(s) on the Critical Line.
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A few pages from G. Chrystal's classic book "Textbook of Algebra" to prove that
Pi is irrational (used on March 8, 2018, in early celebration of Pi Day, that falls during Spring break.)
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Oswald Veblen's beautiful proof that Pi and e are transcendental
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The masterpiece
Pythagorean Primes and Palindromic Continued Fractions,
(that appeared in INTEGERS 5(1) (2005), A30),
by Dr. Z. and mathmagician Arthur Benjamin, that has a combinatorial approach to Henry J.S. Smith's nice proof that every prime that is 1 mod 4 is a sum of
two squares.
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Attempted proof of RH
by Nagy Adel, Mena Asham, and Marian Fady.
(received by Dr. Z.'s on April 1, 2018)
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E.C. Titchmarsh's classic monograph on
the Zeta Function of Riemann (revised by D.R. Heath-Brown)
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Richard P. Brent's paper confirming RH for the first 70 millions zeros of ζ(s)
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Charles F. Van Loan's chariming article about iterating a polygon
that appeared in SIAM News, April 2018 issue.
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Dr. Zeilberger's Office: Hill Center 704
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Dr. Zeilberger's Email: DoronZeil at gmail dot com
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Dr. Zeilberger's Office Hours (Spring 2018): MTh 10:45am-11:30am and by appointment
Description
Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction.
In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards,
and that should be very helpful in whatever mathematical specialty they are doing (or will do) research in.
We will first learn Maple, and how to program with it. This semester we will learn about the Riemann Zeta function,
following the terse, but crystal-clear,
notes
of Professor Henryk Iwaniec, but we will really understand it, since we will program everything,
and the only way to really understand something is to program it.
There is no overlap with previous years.
The prerequisites are the standard courses in real and complex analysis.
In particular, no prior knowledge of Maple, or any programming experience, is assumed.
Also, no overlap with previous years.
Added Feb. 4, 2018: Read Don Zagier's nice exposition of Donald Newman's short proof
of the Prime Number Theory and
Normal Levinson's very lucid Elementary Proof of the Prime Number Theorem .
Optional (but Highly recommended) Getting Started Homework
Teach yourself the basics of Maple by reading Frank Garvan's golden-oldie
part 1,
part 2
Note: a few commands are no longer valid in today's Maple. The
most important one is that " has been replaced by %.
For more details, see this 2002 masterpiece.
How to submit homework
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Every class has a homework assignment. Both Mondays and Thursdays' are due
10:00pm Sunday of the following week.
It should be sent to the email address
ShaloshBEkhad at gmail dot com
Subject: HomeWork#X
and then attach a .txt file(s) called
hwXYourFirstNameYourLastName.txt
where X is the number of the assignment
Except for the first assignment, you should have two attached .txt files.
For example, when Anthony Zaleski submits homework assignments 2 and 3, he should
email ShaloshBEkhad at gmail dot com, with
Subject: HomeWork#2#3
and attach the text files (the source code plus human comments (such lines must start with #)
hw2AnthonyZaleski.txt and hw3AnthonyZaleski.txt
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The very first line of the .txt files should have
#Please do not post homework
or
#OK to post homework
Followed by
#Your Name, Date, Assignment X
Diary and Homework assignments
Programs done on Thurs., Jan. 18, 2018
C1.txt ,
Contains procedures
Homework for Thurs., Jan. 18, 2018, class (due Sunday Jan. 21, 10:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: hw#1
and an attachment
hw1FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
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[People who took this class before, or already know Maple, are exempt from this]
Read and do all the examples, plus make up similar ones,
in the first 30 pages of Frank Garvan's
awesome Maple booklet
(
part 1,
part 2)
Only record a small sample in hw1YourName.txt .
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[For everyone, edited Jan. 21, 2018]. Which of the following sequences
Conv(1,1), Conv(Conv(1,1),1), Conv(Conv(1,1),Conv(1,1))
are already in the On-Line-Encyclopedia of Integer sequences (give their serial number),
and which are not?
Hints:
For the first 20 terms, Conv(1,1), type, in Maple (after you uploaded C1.txt ),
seq(Conv(i->1,i->1,n),n=1..20);
For the first 20 terms, of Conv(Conv(1,1),1), type, in Maple
seq( Conv(i->1, i->Conv(i->1,i->1,i),n),n=1..20);
For the first 20 terms, of Conv(Conv(1,1),Conv(1,1)), type, in Maple
seq( Conv( i->Conv(i->1,i->1,i) , i->Conv(i->1,i->1,i),n),n=1..20);
Then copy-and-paste these sequences to the window of the OEIS, and see what happens.
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[Mandatory for experts, optional for novices]
Write another version of divisors(n), call it DivSmart(n), that first factorizes n, using the command ifactor (or otherwise) , and uses the fact that the set
of divisors of
p1^a1 *....* pk^ak is
the "Cartesian product" of {1,p1,...,p1^a1} x ... x {1,pk,...,pk^ak}
[Hint: you may use a recursive procedure, find the largest prime divisible by n, then the partest power, for which it appears, let's call it p^a,
then apply it recursively to n/p^a and then take the "Cartesian product" of the output with {1, ..., p^a}]
Added Jan. 20, 2018: Note that the "Cartesian product" is in quotes, I meant readly the "set product",
where AxB={ab | a in A, b in B}
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[Optional big challenge, I don't know the answer]
Using Maple (or human ingenuity) , conjecture the answer to Problem 12014 in Jan. 2018 Monthly problems.
[Added Jan. 20, 2018: The convergence is so slow, that it is impossible to conjecture an answer. It looks like
you need to do it the human way. But once you derived the answer, you can check it with Maple, by taking n very large].
[Added Jan. 22, 2018: Edna Jones solved it completely (she got 3 dollars), and Matt Charlney and Yukun Yao solved it partially (they each got one dollar).]
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[Optional Challenge, 5 dollars to be divided among solvers for each of the problems,]
Use Maple to prove as many problems in the
Jan. 2018 Monthly problems.
Added Jan. 22, 2018: Read the nice solutions .
Programs done on Monday Jan. 22, 2018
C2.txt ,
Contains procedures
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Conv(L1,L2), inputs two LISTS L1, L2, and outputs the list that is the convolution
of length min(nops(L1),nops(L2))
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Mo(n): The Mobius function from scratch, using (1.9) in Professor H. Iwaniec's masterpiece
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M(n): the partial sum of Mo(n), the Mertens function
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T(L): (1.11) inputs a sequence L and outputs a sequence of the same length
where the n-th entry is the sum of L[d] over divisors of n
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IT(L): The inverse of T(L)
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phi1(n): phi(n), using the definition that phi(n) is the number of integers <=n coprime with n
Homework for Monday, Jan. 22, 2018, class (due Sunday Jan. 28, 10:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: hw#2
and an attachment
hw2FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
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[People who took this class before, or already know Maple, are exempt from this]
Read and do all the examples, plus make up similar ones,
in pages 30-60 of Frank Garvan's
awesome Maple booklet
(
part 1,
part 2)
Only record a small sample in hw2FirstNameLastName.txt .
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[Mandaotry for experts, optional but recommended to novices; Typo corrected Jan. 23, 2018 (thanks to Edna Jones)]
Write procedure phi2(n) using the formula n*prod(1-1/p, p|n)
and procedure phi3(n) using the formula n*sum(mu(d)/d,d|n). Verify that they are both the same as phi(n)
for n from 1 to 1000
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[Mandaotry for experts, optional but recommended to novices. Modified Jan. 23 (thanks to Edna Jones)]
Let f(s):= abs(Zeta(1/2+s*I)), Find, the smallest zero (on the critical line), to ten decimal digits
Hint: One way is to start with a resolution, say, 0.1, and look at
seq(f(14+0.1*n),n=1..100)
and find the place where it is smallest, then make the resolution smaller, say, 0.001, and explore in the
neighborhood of the previous low place. Etc.
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[Mandaotry for experts, optional but recommended to novices]
Program Equation (1.19) on p. 6 of Iwaniec' book, i.e. write a procedure
Lkn(k,n) that inputs k and n and outputs Λk(n)
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[Mandaotry for experts, optional but recommended to novices]
Verify empirically (1.23) for m,n < 100
Added Jan. 24, 2018: Edna Jones noticed that the above statement is not valid for all m,n, but only if
they satisfy a certain condition between them. What is the condition?
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[Mandaotry for experts, optional but recommended to novices]
Verify empirically (1.24) approximately for s=11, by truncating the infinite sums at 1000, and using evalf.
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[For every one] Do a first reading of Chapter 2 of H. Iwaniec's great book
Added Jan. 29, 2018: Read the nice solutions .
Stuff done on Thurs. Jan. 25, 2018
Today we were fortunate to have a
guest-lecture
by
guru Neil Sloane
Homework for Thurs., Jan. 25, 2018, class (due Sunday Jan. 28, 10:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: hw#3
and an attachment
hw3FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
-
[People who took this class before, or already know Maple, are exempt from this]
Read and do all the examples, plus make up similar ones,
in pages 61-90 of Frank Garvan's
awesome Maple booklet
(
part 1,
part 2)
Only record a small sample in hw3FirstNameLastName.txt .
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[For everyone] Verify (1.1) for n all less than 1000 of the inequality (1.1) in
Jeff Lagarias' nice paper,
make sure Digits is high enough, say 50.
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[Mandaotry for experts, optional but recommended
for novices] Write a Maple program (without peeking in the OEIS entry) that inputs a positive integer n and outputs the
first N terms of sequence A64413
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[Optional Challenge, no deadline]
To do more with coordination sequences (and with no deadline), see
Dr. Sloane's notes on coordination sequences.
Please contact Dr. Sloane directly with any progress.
Added Jan. 29, 2018: Read the nice solutions .
Programs done on Monday Jan. 29, 2018
C4.txt ,
Contains procedures
Homework for Jan. 29, 2018 (due Sunday Feb. 4, 2018, 10:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: hw#4
and an attachment
hw4FirstNameLastName.txt
and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 4
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[People who took this class before, or already know Maple, are exempt from this]
Read and do all the examples, plus make up similar ones,
in pages 90-end of Frank Garvan's
awesome Maple booklet
(
part 1,
part 2)
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[For everyone] Go over Chapter 2, and convince yourself that every step is not hard.
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[For everyone] Find a typo in the second line above Eq. (2.10) on p. 10 of the book
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[Mandatory for experts, optional but recommended for novices]
Write a program
CheckI212(g,x,a,b) ,
that inputs an expression g and a variable x, such that g is a function of x, and real numbers a and b
(such that a < b) and finds the ratio of the left side to the right side. Check that it is indeed ≤ 1 for
g=x^3, a=1, b=5 ; g=exp(x), a=2, b=3 ; g=1/x, a=1, b=4 .
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[Mandatory for experts, optional but recommended for novices.]
Write a program
CheckI213(h,g,x,a,b) ,
that inputs expressions h and g and a variable x, such that h and g are functions of x, and real numbers a and b (a < b) and
finds the ratio of the left side to the right side. Check that it is indeed ≤ 1 for
h=x^2, g=x^3, a=1, b=5 ; h=x^3, g=exp(x), a=2, b=3 ; h=1/x^2, g=1/x, a=1, b=4 .
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[Mandatory for experts, optional but recommended for novices. Corrected Feb. 3, 2018 thanks to Yukun Yao and Matthew Charnely]
Write a program
CheckI218(h,g,x,a,b) ,
that inputs expressions h and g and a variable x, such that h and g are functions of x, and real numbers a and b (a < b) and
finds the ratio of the left side minus the first term of the right hand side divided by H/(1-θ).
You must first first find θ given by equation (2.15)
Check that it is indeed ≤ 1 for
h=x, g=C*x^3, a=1, b=5 ; h=x^4, g=C*exp(x), a=2, b=3 ; h=x^3, g=C*1/x, a=1, b=4 ,
where C is a constant (that you should find) that would make the hypothesis ,(2.15), correct.
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Read and understand Chapter 3.
Added Feb. 5, 2018: Read the nice solutions .
Programs done on Feb. 1, 2018
C5.txt ,
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vM(n): The von-Mangoldt function log(p) is n=pa 0 otherwise
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CheckI18 (N): Checks (1.18) that log(n)=Sum(vM(d), d in divisors of n) for all n from 1 to N
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Sx(x): sum of log(n) from 1 to x
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CheckI24(x):Checks Eq. (2.4)
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psix(x): the famous ψ(x), defined as the sum of log(p) over all prime powers p^a less than x
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CheckI25half(x): cheks that Sx(x) equals the sum of ψ(x/m) for m from 1 to x
Homework for Feb. 1, 2018 (due Sunday Feb. 4, 2018, 10:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: hw#5
and an attachment
hw5FirstNameLastName.txt
and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 5
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Use Maple to verify (both parts) of Problem 12022 in
Feb. 2018 Monthly problems.
for n ≤ 100 (and ALL x except x=1)
[Challenge (5 dollars to be divided): prove it for all n]
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Use Maple to verify Problem 12023 in
Feb. 2018 Monthly problems.
for many randon α s.
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[Challenge (5 dollars to be divided): prove it for all positive real α]
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Write procedures CheckI36(x), CheckI37(x), CheckI38(x), to check Equations (3.6), (3.7), (3.8) respectively in
H. Iwaniec's great book .
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Write a procedure M(x) using (3.9) and write a procedure CheckI311(x) that checks empirically Eq. (3.11)
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Read pp. 15-16 in H. Iwaniec's great book .
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[Optional challenges, open until solved] Prove as many problems as you can from the Feb. 2018 Monthly problems.
Added Feb. 5, 2018: Read the nice solutions .
Programs done on Feb. 5, 2018
C6.txt ,