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 ]

• Time: Mondays and Thursdays , period 3 (12:00noon-1:20pm)

• Main Textbook: "Lectures on the Riemann Zeta Function" by Henryk Iwaniec.
  Other texts used:
  • Donald Newman's short proof by Don Zagier
  • Bernhard Riemman's memoir (English translation) [original,   original mansuscript]
  • 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).
  • George Polya's beautiful elementary proof, scanned from pp. 40-41 of the classic book "A brief introduction to Theta Functions" by Richard Bellman.
  • The first 100000 zeros of Zeta(s), (downloaded from the Andrew Odlyzko's website)
  • 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"
  • 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, p_n in terrms of the elementary symmetric functions, e_n, see p.28 of Ian Macdonald's classic book, "Symmmetric Functions".]
    
  • Chris Houghes' nice article on the moments of the Riemann Zeta function and Random Matrix Theory
  • The wikipedia article on Hough Montgomery's Pair Correlation Conjecture.
  • Atle Selberg's seminal paper on the zeros of Zeta(s) on the Critical Line.
  • 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.)
  • Oswald Veblen's beautiful proof that Pi and e are transcendental
  • 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.
  • Attempted proof of RH by Nagy Adel, Mena Asham, and Marian Fady. (received by Dr. Z.'s on April 1, 2018)
  • E.C. Titchmarsh's classic monograph on the Zeta Function of Riemann (revised by D.R. Heath-Brown)
  • Richard P. Brent's paper confirming RH for the first 70 millions zeros of ζ(s)
  • Charles F. Van Loan's chariming article about iterating a polygon that appeared in SIAM News, April 2018 issue.

• Dr. Zeilberger's Office: Hill Center 704
• Dr. Zeilberger's Email: DoronZeil at gmail dot com
  
• 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

For more details, see this 2002 masterpiece.

How to submit homework

• 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

• 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

• DiriF(f,s,N): inputs a number-theoretical function f and a symbol s and a positive integer N outputs the partial sum up to N of the Dirichlet series of f(n)

• DumbDiv(n): a home-made version of divisors(n) (set of divisors of the pos. integer n)

• YukunDiv(n): a smarter home-made version of divisors(n) (set of divisors of the pos. integer n)

• Conv(f,g,n)

Homework for Thurs., Jan. 18, 2018, class (due Sunday Jan. 21, 10:00pm)

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.

1. [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 .

2. [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.

3. [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}

4. [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).]

5. [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

• Conv(L1,L2), inputs two LISTS L1, L2, and outputs the list that is the convolution of length min(nops(L1),nops(L2))

• Mo(n): The Mobius function from scratch, using (1.9) in Professor H. Iwaniec's masterpiece

• M(n): the partial sum of Mo(n), the Mertens function

• 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

• IT(L): The inverse of T(L)

• 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)

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.

1. [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 .

2. [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

3. [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.

4. [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 Λ_k(n)

5. [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?

6. [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.

7. [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)

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.

1. [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 .

2. [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.

3. [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

4. [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

• CheckI23(f,n,x): Checks Eq. (2.3) in H. Iwaniec's book

• CheckI27(s1,L): Checks (2.7) for numeric s and pos. integer L

• Dir(x): (2.9) sum(tau(n),n=1..x):

• CheckI210(x): checks (2.10)

Homework for Jan. 29, 2018 (due Sunday Feb. 4, 2018, 10:00pm)

Subject: hw#4

and an attachment

hw4FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 4

1. [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)

2. [For everyone] Go over Chapter 2, and convince yourself that every step is not hard.

3. [For everyone] Find a typo in the second line above Eq. (2.10) on p. 10 of the book

4. [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   .

5. [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   .

6. [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.

7. Read and understand Chapter 3.

Added Feb. 5, 2018: Read the nice solutions   .

Programs done on Feb. 1, 2018

• vM(n): The von-Mangoldt function log(p) is n=p^a 0 otherwise

• CheckI18 (N): Checks (1.18) that log(n)=Sum(vM(d), d in divisors of n) for all n from 1 to N

• Sx(x): sum of log(n) from 1 to x

• CheckI24(x):Checks Eq. (2.4)

• psix(x): the famous ψ(x), defined as the sum of log(p) over all prime powers p^a less than x

• 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)

Subject: hw#5

and an attachment

hw5FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 5

1. 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]

2. Use Maple to verify Problem 12023 in Feb. 2018 Monthly problems. for many randon α s.

3. [Challenge (5 dollars to be divided): prove it for all positive real α]

4. Write procedures CheckI36(x), CheckI37(x), CheckI38(x), to check Equations (3.6), (3.7), (3.8) respectively in H. Iwaniec's great book .

5. Write a procedure M(x) using (3.9) and write a procedure CheckI311(x) that checks empirically Eq. (3.11)

6. Read pp. 15-16 in H. Iwaniec's great book .

7. [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

• Z(s,N)

• Ph(s,N): sum(log(p)/p^s, first N primes)

• th(x): sum(log(p), p<=x)

• EP(s,N): the product of 1/(1-1/p^s)) over the first N primes

• CheckZ4(N,alpha,eps)

• CheckZ5(N): checks (V) in Zagier's paper (under construction)

Homework for Feb. 5, 2018 (due Sunday Feb. 11, 2018, 10:00pm)

Subject: hw#6

and an attachment

hw6FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 6

1. [typo corrected, Feb. 9, 2018, thanks to Edna Jones] Get procedure CheckZ5(N) to work by first creating a list of floating-point numbers for (θ(x)-x)/x^2 from 2 to N and then adding them, i.e. replacing integration by summation

2. Read, and completely understand the proof in Don Zagier's nice exposition of Donald Newman's short proof

3. Write a procedure

   gT(f,t,z,T)

   tha inputs an expression f in a variable t, a complex number z, and a real number T, and outputs the integral of f(t)*exp(-z*t) for t rom 0 to T. Experiment with several f and see for which ones the limit of g_T(0), at T goes to infinity, seems to converge, and whether the limit seems to go to g(0).

4. [Optional Challenge] Empiriclly (or better still, rigorously) conjecture (or determine) answers to the limits in problem 12026 in Feb. 2018 Monthly problems.

   [Note added Feb. 8, 2018: the convergence is extremely slow, but Professor Kauers figured out the answer to the first limit, using ingenuity and symbolic computation. Neither of us knows the answer to the second limit. Update Feb. 9, 2018: Prof. Kauers also figured out the answer to the second limit, suprisingly, the first limit is very simple, and the second limit is rather complicated. Can you find them?]

Added Feb. 12, 2018: Read the nice solutions   .

Programs done on Feb. 8, 2018

Homework for Feb. 8, 2018 (due Sunday Feb. 11, 2018, 10:00pm)

Subject: hw#7

and an attachment

hw7FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 7

1. Inspired by Professor Kauers' guest lecture, but ising Maple, conjecture a linear recurrence operator annihilating the sequece
   sum(binomial(2*n,n+k)*binomial(n,k)^2,k=0..n)
   (where N is the shift operator in n (for example the fiboancci numbers are annihilated by N^2-N-1 and n! by N-(n+1)). You may use procedure Findrec in Dr. Z.'s Maple package FindRec.txt
   (Type ezra(Findrecc) for instructions how to use it).

2. Having found the operator, try to use Dr. Z.'s Maple package AsyRec.txt to figure out the asympotic behavior of the above sequence (Use procedure Asy, or better still, AsyC).

Added Feb. 12, 2018: Read the nice solutions   .

Programs done on Feb. 12, 2018

• xi(s): the left side of the FE on p. 21 in H. Iwaneic's book, in other words the symmetrized version of the Zeta function

• ZZ(): the list of the first 177 zeros of Zeta(s)

• Check5C(n), where T is the n-th zero of Zeta(s)

Special Homework for Feb. 12, 2018 (due Wed., Feb. 14, 2018, 6:00pm)

Subject: hw#8special

and an attachment

hw8specialFirstNameLastName.jpeg , or hw8specialFirstNameLastName.png , or hw8specialFirstNameLastName.gif

and indicate whether it is OK to post (in the body, not the attachment)

• By using the plots package (type: with(plots); ), and procedures textplot and (if you wish) polarplot (or other commands, even plot3D), and lots of nice colors, create as nice a Valentine as you can, whose inside says "I Love Experimental Mathematics". Export is as a .png or .jpeg or .gif file (you may have to look up how to do it).

  [Hint: one possibility is the polar plot of r=arccos(sin(θ)), make sure to have the option axes=none, i.e. polarplot(arccos(sin(t)),t=0..2*Pi, axes=none)]

  you are welcome to search the internest for nicer mathematical valentines, some with 3D. The students will vote who is the nicest, and the winner witll win a chocolate valentine.

Added Feb. 14, 2018: Admire the beautiful valentines designed by the students, using Maple.

Usual Homework for Feb. 12, 2018 (due Sunday Feb. 18, 2018, 10:00pm)

Subject: hw#8

and an attachment

hw8FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 8

1. Using the plots package, write a procedure,Contour(R, delta) that inputs a (large) positive real number R, and a small positive integer delta, and draws the the curve C in the proof of the analytical theorem in Don Zagier's paper (p. 707)

2. Write a procedure, Check5D(s,N), that inputs a complex number s and a positive integer N less than 177, and checks, numerically Fact D in H. Iwaniec's book (p. 21) where the product is taken over the first N roots, by taking the ratio of the left side to the right side, and hopefully getting something close to 1.

   What are: Check5D(3+I,10), Check5D(13+I,100) ?

3. Write a procedure, Check5E(x,N), that inputs a large real number, x and a positive integer N less than 177, and checks, numerically, fact E, where the sum is over the first N roots of Zeta(s). (See (10.1), p. 37 for the meaning of the "sigma flat"), by taking the difference between the left side and the right side, and hopefully getting something close to 0

4. Read and understand Chapter 6 (pp. 23-24)

Added Feb. 19, 2018: Read the nice solutions   .

Programs done on Feb. 15, 2018

• Th(t,N): The truncated Jacobi theta function (with one variable)

• CheckJac(t,N): checks numerically Jacobi's identity θ(1/t)=sqrt(t)θ(t)

• A faster way (for small t) to compute Th(t,N) using Jaocbi's identity

• CheckTay(p,x): Checks the correctness of Taylor's expansion (at x=0) for polynomials p of x

• RaTP(x,N): a random trig. polynomial of period 1 of degree N with integer coefficients less than 1001

• FS(f,x,N): The Fourier series of a function f of x on [0,1], truncated to N

• [modified Feb. 18, 2018, thanks to Matt Charnley] FSn(f,x,N,x0): The Fourier series of a function f of x on [0,1], evaluated at x0, truncated to N,done numerically, should be close (for large N) to subs(x=x0,f);

Homework for Feb. 15, 2018 (due Sunday Feb. 18, 2018, 10:00pm)

Subject: hw#9

and an attachment

hw9FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 9

1. [Modified Feb. 18, 2018, thanks to Matthew Charnley, who won a dollar.]
   Write a procedure

   FindN(f,x,k,x0)

   that inputs an explicit function f in the variable x, (for example) that is obviously continuos on the open interval (0,1), and a positive integer k, as well as a number x0 strictly between 0 and 1 (i.e. in the open interval (0,1)) and outputs the smallest N such that abs(FSn(f,x,N,x0)-subs(x=x0,f)) is less than 1/10^k, in other words what N should you take to guarantee k decimal digits. What are (take Digits:=30)

   FindN(1/(1+x)^3,x,3,0.5) , FindN(sin(cos(x)),x,3,0.8) , FindN(abs(x-1/2),x,3,0.6), FindN(abs(x-1/2),x,3,0.51), , FindN(abs(x-1/2),x,3,0.501)

2. Read and understand these two pages, proving the identity that is crucial for the proof of the functional equation for Zeta(s)
   (Note the last paragraph of the proof is in the next page, that is not scanned, but it is the end of the standard proof that int(exp(-x^2),x=-infinity..infinity) is sqrt(2*Pi) that you should know from Calc3. (If I=int(exp(-x^2),x=-infinity..infinity) then I^2=int(exp(-x^2),x=-infinity..infinity)*int(exp(-y^2),x=-infinity..infinity)= int(int(exp(-x^2-y^2),x=-infinity..infinity),y=-infinity..infinity), now convert to polar coordinates)

3. Read and understand Chapter 7 in H. Iwaniec's book

4. Write a procedure

   Check82(f,R)

   that checks Lemma 8.2 (p. 28) In H.Iwaniec's book for a holomorphic function f(z) and a positive real number R. Try it out for

   Check82(exp(z)*mul((z-i),i=1..5),R);

   for R=1,R=2,R=3,R=4,R=5, and R=7

5. Do a first reading of Chapter 8

Added Feb. 19, 2018: Read the nice solutions   .

Programs done on Feb. 19, 2018

• sin1(z,N): the partial product for Euler's product formula for sin(z)

• Check92(n): checks (9.2) with T=the n-th root of Zeta(s)

• Check121(x,y,sig): checks (12.1) in H. Iwaniec's book

Homework for Feb. 19, 2018 (due Sunday Feb. 25, 2018, 10:00pm)

Subject: hw#10

and an attachment

hw10FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 10

1. Read and understand Euler's proof that Zeta(2)=pi^2/6 as nicely told in Brendan Sullivan's paper Numerous Proofs that ζ(2)=π²/2. Try to read and understand the proof for Zeta(2n). Also read and understand Apostol's proof for Zeta(2).

2. Write a procedure Check127(t,x,N) that inputs a real number t and a real number x (not far from sqrt(sqrt(1/4+t^2)/(2*pi)) and checks Eq. 12.7, where the infinite sums are replaced by sums from n=1 to n=N. Test the procedure with some reasonable inputs.

3. Recall the Newton-Raphson method (that also works for complex functions) that approximately finds a root of an equation P(z)=0 by starting with an initial guess z0, and replaces it by z0-P(z0)/P'(z0). Write a procedure

   NR(f,z,z0,eps)

   that inputs an expression (representing a function) f in the variable z, a complex number z0, and a small positive real number eps, and outputs an approximate root obtained by iterating the above until the difference in absolute value of two consecutive iterations is less than eps.

   Test it with several random polynomials and initial values with eps=1/10⁸.

4. By using NR(Zeta(s),s, 1/2+I*n, 1/10^7), for integer n (starting with n=10) try to find, from scratch as many zeros of Zeta(s) that you can. Compare it with the Odlyzko table.
   Comment added Feb. 28, 2018: Edna Jones made the interesting comment that the reason NR does not converge to the nearest zero is because (even for a cubic!) the beahvior is "chaotic" and the "basins of attraction" are fractals. See wikipedia article .

Added Feb. 26, 2018: Read the nice solutions   .

Programs done on Feb. 22, 2018

Contains

• PfE(inputs a list of the [e1,...en] and outputs p_n (implementing the det. expression in Ian Macdoland's famous book Symmetric Polynomials (p. 28))

• DP(L1,L2,M): inputs the first nops(L1) terms of a Dirichlet polynomial (or series),P1 and similarly for L2, P2, outputs the first M terms of the product P1*P2

• DerD(L,k): the kth derivative of the Dirichlet polynomial given by the list L

• Dpo(L,k,N):the first N terms of the Dirichlet poly L^k

Homework for Feb. 22, 2018 (due Sunday Feb. 25, 2018, 10:00pm)

Subject: hw#11

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#YourName, Math 640 (Spring 2018) homework assignment 11

1. Read and understand Dr. Z's gorgeous combinatorial proof of Newton's identitities and find the typo(s) [there is at least one], and correct it (them).

2. Write a Maple procdeure T(A,j,l) that implements the bijection, make sure that it is an involution. Also write a program Wt(A,j,l,x) that outputs the weight.

3. Use Procedure PfE() to write a procedure ZetaEven(n) that inputs a positive integer and outputs Zeta(2n)

4. Try to see whether PfE([seq(1/(2*i)!,i=1..n)]), PfE([seq(1/(3*i+1)!,i=1..n)]) etc produce interesting sequences of rational numbers.

5. Write a procedure LogDerD(L,n) that inputs the first nops(L) coefficients of a Dirichlet polynomial A(s), and outputs -A'(s)/A(s) (Eq. (13.8) in Iwaniec's book.)

6. Write a procedure Check1312(L,T) that checks Theorem 13.1 (Eq. 13.12) for a Dirichlet polynomial, of degree nops(L), given by the list L. Try it out for several random L and (large) T.

7. Write a procedure Check1318(T) that checks Eq. (13.18) (p. 50) of H. Iwaniec's book, by bounding the "implied constant". How big is it?

8. Write a procedure Check1320(T) that checks Eq. (13.20) (p. 51) of H. Iwaniec's book, by bounding the "implied constant". How big is it?

9. [Optional challenge, I have no clue, 5 dollars to be divided] Explain the inequality sign going from line 4 of p. 50 to line 5 of p.50 in Professor Henryk Inwaniec's book "Lectures on the Riemann Zeta Function" .
   Added Feb. 26, 2018: Professor Iwaniec confirmed that it was an error, see the corrected page . [Thanks to Edna Jones]

Added Feb. 26, 2018: Read the nice solutions   .

Programs done on Feb. 26, 2018

Contains

• Mk(k,T):The k-th moment of Zeta(1/2+i*t) from t=0 to t=T divided by T (Warning: very slow)

• OurInt(f,x,a,b,N): Numerical integration of the function of x given by the expression f from x=a to x=b, using the very naive midpoint rule, y dividing it into N rectangles

• MkF(k,T,N):The k-th moment of Zeta(1/2+i*t) from t=0 to t=T divided by T, done by using the very naive Midpoint rule

• deltn(n):The spacing between the (n+1)-th non-trivial and gamma_n:=n-th non--trivial of Zeta(n) times log(gamma[n]/(2*Pi)) /(2*Pi). It features in Hugh Montgomery' conjecture

• HM(N,K): the list of partial sums delt(n)+...+delt(n+k) for k from 1 to K

Homework for Feb. 26, 2018 (due Sunday March 4, 2018, 10:00pm)

Subject: hw#12

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#YourName, Math 640 (Spring 2018) homework assignment 12

1. Write procedures TrapInt(f,x,a,b,N) and SimpsInt(f,x,a,b,N) for numerical integration

2. [Rephrased Feb. 28, 2018 (thanks to Edna Jones)]
   Write procedures a(k,eps) that approximates a(k) by truncating the infinite product and sums (stopping them, at say the N-th prime and then the 2N-th prime and making sure that their difference is less than eps) For the definition of a(k) see Chris Houghes' nice slides on the moments of the Riemann Zeta function and Random Matrix Theory,

3. Using SimpsInt(f,x,a,b,N) for sufficiently large N, and the above a(k,eps), verify empirically the The Keating-Snaith conjecture given on p.13 of Chris Houges' slides, by writing procedure

   KS(T,k)

   that computes the ratio of the left side to the right side.
   [Note: G(k) (k ≥ 2) is nothing but "iterated-factorial": 0!*1!*...*(k-2)! and G(1)=1, see wikipedia, thanks to Edna Jones for correcting a previous error]

   What are (approximations of) KS(100,3), KS(1000,3); KS(100,3), KS(100,4) ?

4. [corrected March 4, 2018, thanks to Edna Jones]
   Write a procedure

   VerifyHoughesP18(T,k) that computes the ratio of the left side to the right side in the conjecture on top of p.18 of Chris Houges' slides. What is the values of VerifyHoughesP18(10000,2)?

5. [Rephrased Feb. 28, 2018 (thanks to Edna Jones)]
   Write a procedure

   VerifyGonekConj(T)   ,

   that computes the ratio of the left side to the right side in Gonek's conjecture given on the bottom of p.18 of Chris Houges' slides. are the values of (keep trying until it takes too long) VerifyGonek(T) for

   T=100, T=1000, T=10000, T=100000, etc.

Added March 5 2018: Read the nice solutions   .

Programs done on March 1, 2018

Contains

• H(s): Eq. (15.3) (the factor needed in the Functional Equation of Zeta(s))

• Z(u): the nomrmalized Zeta function for s=1/2+I*u, Eq. (15.2)

• It(t,D1): Eq. (15.5)

• Jt(t,D1): Eq. (15.6)

• CheckP58L10(): checks Line 10 p. 58 of Iwaniec's book

Homework for March 1, 2018 (due Sunday March 4, 2018, 10:00pm)

Subject: hw#13

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#YourName, Math 640 (Spring 2018) homework assignment 13

1. Read and understand the bottom of p. 58 and top of p. 59. Explain why he set S contains a sequence {t1, ..., tR} of Δ-spaced points with R ≥ T/(2Δ)

2. Read and try to understand the proof of Lemma 15.2, by explaining every line. (I know it is tough going!)

3. Write a program

   CheckP59L23()   ,

   that checks Line 23, p. 59 of Iwaniec's book by estimating (i.e. finding an upper bound) , for large T and t, and any u, the implied constant in the O((u^2+1)/T)

4. Do a first reading of Chapters 16 and 17 of H. Iwaniec's book .

Added March 5 2018: Read the nice solutions   .

Programs done on March 5, 2018

Contains

• H(s) 1/2*s*(1-s)*Pi^(-s/2)*GAMMA(s/2)

• xi(s) H(s)*Zeta(s)

• LogHs(s): the log. der. of H(s), H'(s)/H(s)

• Y(g,s) : Eq. (17.4) in Iwaniec' book

• G(g,s): Zeta(s)+g*diff(Zeta(s),s), Eq. 17.5

• Check174(g,s0): checks Eq. (17.4) with G(s) and Y(s) given by (17.5) and (17.6)

• Check187(s,m): the left side of (18.7) divided by the right side for any s=s0 and m

• Check188(s0,j,T)

Special Homework for March 5, 2018 (due March 7, 2018, 10:00pm, In Early celebration of Pi Day)

Subject: hw#14Pi

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#YourName, Math 640 (Spring 2018) homework assignment 14Pi

1. Using Maple, simulate a program, Buffon(K), that approximates Pi by throwing a Buffon needle K times. What did you get for Buffon(1000)? Buffon(10000)?

   Hint: in order to generate a random location for the (x-coordinate of the) bottom of the needle use
   evalf(RandomTools[Generate](positive)):
   and in order to generate a random orientation (i.e. the angle it makes with the positive x-axis) for the needle, use
   evalf(RandomTools[Generate](positive)*2*Pi):
   now test whether the x-coordinate of the top of the needle is either negative or larger than 1. Now repeat this many times and find the ratio of successes, let's call it r, that should be an approximation for 2/Pi, so to estimate Pi, à la Buffon, compute 2/r.

2. Using the fact that binomial(2n,n)/2^(2*n) is approximately (for large n) , 1/(sqrt(n*Pi), write a Maple program

   CoinTossing(n,K)

   where each round consists of throwing a fair coin 2*n times, and the round is considered a susccess if there were exactly n Heads. Let S be the number of successful rounds. Use the fact that for large n and K, S/K is approximately 1/sqrt(n*Pi), to estimate Pi. What did you get for CoinTossing(100,10000)? Do it several times.

3. [Contest, prize is a Pi-shaped chocolate piece] Write a simulation program that is even slower than the above, i.e. find something whose probability can be expressed in terms of Pi, simlate it, using Maple's rand(), and use it to approximate Pi, as slowly and as poorly as possible, BUT, if you had all of the time in the world, and believe in the law of large numbers, you can get it to any desired accuracy.

Added March 8 2018: Read the nice solutions   . The co-winners of the prize (a Pi-shaped chocolate piece) are Yukun Yao and George Hauser.

Regular Homework for March 5, 2018 (due Sunday March 18, 2018, 10:00pm)

Subject: hw#14

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#YourName, Math 640 (Spring 2018) homework assignment 14

[Rephrased March 7, 2018 (thanks to Edna Jones)]

1. Write a procedure

   Q(x,T,gL)

   that inputs a real numbre x a (usually) large positive real number T and a finite list of real numberss gL and outputs the expression, Q(x), defined in Eq. (18.12), in Professor Iwaniec's great book , where g_1=gL[1], g_3=gL[2], ...,

2. Write procedure del(s,T) defined by (18.11) and G(s,T,gL) defined by 18.10

3. Read Chapter 19.

4. Use Maple to plot a figure similar to Fig. 19.1 on p. 71.

5. Browse Atle Selberg's seminal paper on the zeros of Zeta(s) on the Critical Line.

Added March 19 2018: Read the students' nice solutions   .

Programs done on March 8, 2018

Contains

• CFI(A,B): the non-simple finite continued fraction with numerators B and denominators by A, see Chrystal's book p. 512, I

• CFII(A,B): the non-simple finite continued fraction with numerators B and denominators by A, with Minus signs see Chrystal's book p. 512, II

• tanRat(c,d,N): the pair A,B that you get when you replace x by c/d , c,d integers (corrected by Edna Jones)

• PiRat(n): rational approximations of Pi better than 22/7

Homework for March 8, 2018 (due Sunday March 18, 2018, 10:00pm)

Subject: hw#15

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#YourName, Math 640 (Spring 2018) homework assignment 15

Read and understand the proof that if c/d is rational then tan(c/d) is irrational, and hence Pi/4 (and hence Pi) are irrational, as proved here .
1. Read and understand Oswald Veblen's beautiful proof that Pi and e are transcendental

2. Use the two formulas in Jesus Guillera's great website to write programs JG1(N), JG2(N) that input a positive integer N and compute Pi to N decimal digits.

3. Use procedure AZd in the Maple package EKHAD.txt to compute

   int( x^n*(1-x)^n/(1+x^2),x=0..1):

   for n from 2000 to 2010.

   Hint: AZd(x^n*(1-x)^n/(1+x^2),x,n,N)[1] will give you a linear recurrence operator that implies a second-order recurrence. Program it with "option remember" and get the 2000-th through 2010-th terms.

4. Look up Machin-type formulas and implement them to compute Pi.

Added March 19 2018: Read the students' nice solutions   .

Programs done on March 19, 2018

Contains

• CIc(f,z,C0,r): the contour integral of f of z over the circle center C0 (a complex number) and radius r #e.g. CIr(1/z,z,0,3);

• CIcN(f,z,C0,r): Numerical version of the above, using numerical integration

• CIr(f,z,C0,a,b): the contour integral of f of z over the rectangle whose bottom-left corner is the point C0=[x0,y0], and horiz. side a and vertical side b, in other words the rectangle whose vertices are [x0,y0], [x0+a,y0],[x0+a,y0+b],[x0,y0+b]

• CIrN(f,z,C0,a,b): Numerical version of the above

Homework for March 19, 2018 (due Sunday March 25, 2018, 10:00pm)

Subject: hw#16

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#YourName, Math 640 (Spring 2018) homework assignment 16

1. Wrie a procedure

   CIpara(f,z,P,t,alpha,beta): that inputs a function f of a complex variable z, a pair P=[P1,P2] where P1 and P2 are functions of the real variable t, and real numbers alpha,beta and outputs the contour integral

   Int(f(z) dz, z in C from start to finish),

   where C is the parametric curve x=P1(t),y=P2(t) from t=alpha to t=beta (Hint: dz=dx+I*dy, dx=P1'(t)dt, dy=P2'(t))

   Make sure that CIpara(f,z,[c0+r*cos(t),c1+r*sin(t)] ,t,0,2*Pi) gives the same output as CIc(f,z,c0+I*c1,r) for various functions f of z

2. By writing a short procedure LineSegment(A,B,t) that outputs the parametric representation of a line segment joining point A=[A1,A2] and point B=[B1,B2] from t=0 to t=1, write a procedure

   CIpolygon(f,z,Polygon): that inputs a function f of a complex variable z, and a list of vertices denoting in order, the vertices of a polygon, and outputs

   Int(f(z) dz, z in C from start to finish), where C is the polygon. Make sure that

   CIpolygon(f,z,[[C0,C1], [C0+a,C1],[C0+a,C1+b],[C0,C1+b]] )

   outputs the same as CIrN(f,z,[C0,C1], a,b) for various functions f of z.

3. Write a procedure

   IntCauchy(L,x)

   that inputs a list L of positive numbers L and computes

   int(1/mul(x^2+L[i]^2,i=1..nops(L)),x=-infinity..infinity) using Cauchy's residue theorem.

   [Hint: Maple has a command called residue. The integral is the same as the contour integral over the contour consisting of the real axis and the upper infinite semicircle. The answer is 2*Pi*I times the sum of the residues of the poloes of the above function in the upper half-plane. Make sure that you get the same answer for various random lists L.

4. Read and understand Chapter 19 of Professor Iwaniec's great book ,

5. Conjecture the answer to March 2018 Monthly Problem 12029, and if possible prove your conjecture.

6. Check numerically March 2018 Monthly Problem 12031, (both parts) and if possible, prove them.

7. Conjecture the answer to March 2018 Monthly Problem 12032, and if possible, prove it, and it would be really nice if you can use procedure MultiZeil in this amazing Maple package
   [Hint, in order to use MultiZeil, you need to divide the summand by 2^n, and calling the double-sum of the problem A(n), get a recurrence for A(n)/2^n, that easily translates to a recurrence for A(n). Note that N is the shift operator in n, so, for example, the recurrence for Fibonacci is abbreviated N^2-N-1. Once you get a recurrence, you can either do it by hand (using Discrete Math 101), or use Maple's command rsolve].

Added March 26 2018: Read the students' nice solutions   .

Programs done on March 22, 2018

Contains

• H(s)

• xi(s) : Zeta(s)*H(s): end:

• diff1(f,x,j): the j-th derivative of f w.r.t. to

• G(s,g): Conrey's function G(s) defined by Eq. (18.1) , where g is a finite list of [g1,g3,g5,...]

• Check185(s,g): checks (18.5) for s and a list g

Homework for March 22, 2018 (due Sunday March 25, 2018, 10:00pm)

Subject: hw#17

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#YourName, Math 640 (Spring 2018) homework assignment 17

1. Write a program

   Check211(S,C0,a,b)

   that checks the Littlewood lemma (Eq. 21.1, p. 77), where S is a finite set of complex numbers, C0 is a point denoting the bottom-left corner of a rectangle whose horizonal side has length a and vertical side has length b, and taking F to be the function of z that is the product of (z-alpha) over all the members alpha of S. You may use CIr(f,z,C0,a,b) or CIrN(f,z,C0,a,b) of C16.txt . Test it for some examples where some membes of S lie inside the rectangle, and some don't.

2. Using the amazing Maple package RENE.txt, prove Problem 12027 of March 2018 Monthly.

   [Hint: Use the triangle Te(m,n), and procedures Inradius(m,n), Circumradius(Te(m,n)), Incircle(m,n) (for the equation of the incircle), Pt to find the intersection of two lines, and Le to find the line joining two points, DistSq (for the square of the distance between two points), and SSR(A,n,B), with n=3, A the list of the squares of the two terms on the left, and B the right side (R/r-1/2).

3. Test empirically Problem 12033 of March 2018 Monthly, and if possible, prove it.

Added March 26 2018: Read the students' nice solutions   .

Programs done on March 26, 2018

Contains

• RatToCF(x): converts a pos. rational number x into a continued fraction

• NuToCF(x,n): appx. a pos. number x into a continued fraction with <=n terms

• EvalCF(L): evaluates the simple continued given by a list L

• SymCF: the rational function of a[1], ..., a[n] describing EvalCF([a[1], ..., a[n]]): the numerator followed by the denominator

• P(a,n): The a-analog of the n-th Fibonacci number, the numerator of SymCF([a[1], ..., a[n]])

Homework for March 26, 2018 (due Sunday April 1, 2018, 10:00pm)

Subject: hw#18

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#YourName, Math 640 (Spring 2018) homework assignment 18

1. Write a program

   FindPeriod1(L,d)

   that inputs a list L and a positive integer d and finds (if they exist), lists L1,L2, such that L is the list L1 followed by the list L2 repeated d times, and the rest is the start of L2. For example

   FindPeriod1([1,2,1,3,2,1,3,2,1,3,2,1,3], 3);

   should output [1,2], [1,3,2] .

2. Using FindPeriod1(L,d), Write a program

   FindPeriod(L)

   that finds a list L1 and L2 for which there exists a d>=3 such that FindPeriod1(L,d)=[L1,L2]. For example FindPeriod([1,2,1,3,2,1,3,2,1,3,2,1,3]) should return [1],[2,1,3] [I thank Yukun Yao for correcting this, before I wrote [1,2],[1,3,2] that is not mininal in the prefix]

3. It is a famous theorem in Number Theory that every quadratic irrationality has an ultimately periodic continued fraction. By using (make DIgits large enough) NuToCF(evalf(x),n) followed by FindPeriod(L), write a procedure

   ConjCF(x)

   that inputs a quadratic irrationality, and outputs lists of positive integers, L1 and L2, such that the infinite (simple) continued fraction of x is L1 followed by L2 repeated infinitely many times.

4. Write a procedure

   EvalPCF(L1,L2)

   that inputs lists L1 and L2 (L1 may be empty) and finds the quadratic irrationality represented by it. For example

   EvalPCF([1],[2]);

   should output sqrt(2)

   [Hint: First write a procedure for the pure-periodic z=L2^infinity , using the fact that z=EvalCF([L2,z]), solve for z, and then do EvalCF([L1,z]).

5. Using ConjCF(x) and EvalCF(L1,L2), write a procedure

   ConjAndProveCF(x)

   that has the same input and output as ConjCF(x) but now the conjectured infinite (ultimately periodic) (simple) continued fraction is rigorously proved (or returns FAIL, if unable to prove it)

6. Find the L1 and L2 for all square-roots of positive integers (that are not perfect squares) from 2 to 100 (and if possible even further). Is the sequence of lengths of the periodic part (L2) in the OEIS? What square-root gives the largest period?

Added April 2, 2018: Read the students' nice solutions   .

Programs done on March 29, 2018

Contains

• SofS(n): the set of all the pairs [a,b], a<=b, such that a^2+b^2=n

• P(L):The numerator of the (simple) continued fraction L

• Q(L):The denominator of the (simple) continued fraction L

• FibS(n): the set of vectors in {1,2} that add-up to n

• wt(s): the weight of the Fibonacci word s, product of a[j] over all intermediate stops j that follow immediately j-1

  CheckQI(L,k): Checks the identity P([a1,...., an])=P([a1...,ak])*P([a(k+1)... an])+ P([a1...,a(k-1)])*P([a(k+2)... an])

Homework for March 29, 2018 (due Sunday April 1, 2018, 10:00pm)

Subject: hw#19

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#YourName, Math 640 (Spring 2018) homework assignment 19

1. Write a program

   SofP(n,k,r)

   that inputs a positive integer n and outputs the set of tuples [a1, ..., ak] of non-negative integers, such that a1>=a2>=..>=ak>=0 and

   a1^r+ ...+ak^r=n

   Verify that SofP(n,4,2) is non-empty for all n<=1000, thereby confirming Lagrange's theorem for n <=1000.

   Find the set of primes p less than 1000 for which SofP(p,3,2) is non-empty. Can you characterize it?

2. In preparation to Henry Smith's proof given a beautiful combinatorial formulation here,write a program

   AllCF(p)

   That inputs a prime p and outputs the set of (simple) continued-fraction representation of the fractions p/a for a from 1 to (p-1)/2. Verify, for all primes less than 1000, that whenver a list shows up, its reverse also shows up.

Added April 2, 2018: Read the students' nice solutions   .

Programs done on April 2, 2018

Contains, in addition to the contents of C18.txt and C19.txt, the following procedures

• HS(p): the set of all the continued fractions of p/i for i from 2 to (p-1)/2

• Rev(L): the reverse of the list L

• HJS(p): finding [a,b] such that p=a^2+b^2 following Henry Smith

• SmithTable(N) : the list of length N of [p,[a,b]] where p=a^2+b^2 and p is 1 mod 4 for the first such primes 1 mod 4, via Henry Smith's algorithm

• SmithTableF(N): Same as SmithTable(N), but done directly

• [Corrected after class]
  EvalCFg(A,B): evaluates the non-simple continued fraction with numerators B and denominators A as given in Chrystal's book (p. 512, Eq. (1))

Homework for April 2, 2018 (due Sunday April 8, 2018, 10:00pm)

Subject: hw#20

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#YourName, Math 640 (Spring 2018) homework assignment 20

1. Check the corrected version of EvalCFg(A,B) and verify that

   EvalCFg([seq(a[i],i=1..n)],[1$n])

   gives the same thing as

   1/EvalCF([seq(a[i],i=1..n)]);

   for n between 1 and 10. Also prove is in general, by looking at the definition.

2. Let Qg(A,B) be the denominator of EvalCFg(A,B), find the natural generalization of wt(s,a) from C19.txt, call it wtG(s,a,b), such that Qg(A,B) is the sum of wtG(s,a,b) of all the members of FibS(n)

3. Let Pg(A,B) be the numerator of EvalCFg(A,B)), express it in terms of Qg(A',B') for some related lists A', B'.

4. Using the combinatorial interpration, find a second order recurrence satisfied by Qg(A,B)

5. By using Maple's

   convert(tan(x),confrac,20);

   find the typo in Eq. (15) (p. 522) of Chrystal's book [Or convince me that it is not a typo, but I misunderstood the notation]

6. [Optional Challenge, 5 dollars to be divided among correct solutions] Define the sequence of polynomials

   Pn:=(n,x)->denom(EvalCFg([seq(2*i-1,i=1..n)],[x,x^2$(n-1)]))

   conjecture a closed form expression for the coefficient of x^k in Pn(n,x), and hence conjecture an explicit expression for Pn(n,x). Then use this to find an expression for Qn(n,x) (the numerator), and finally prove that the limit of Qn(n,x)/Pn(n,x) as n goes to infinity, converges to tanh(x)

7. Write a program

   Verify(a,b,c,d,e)

   that empirically verifies theorem 1 of Marcin Mazur's paper (Amer. Math. Monthly, March 2018). Assume that the vertices of the tetrahedron are

   A=[0,0,0], B=[1,0,0], C=[a,b,0], D=[c,d,e]

   and hence its volume is b*e/6.

8. [Optional Challenge, 5 dollars to be divided among correct solutions] Prove this Theorem using Maple.

9. Using Maple, prove completely Proposition 2 of Marcin Mazur's paper (Amer. Math. Monthly, March 2018).
   [Take A,B,C,D as above, look up the definition of the centroid, and let P be an arbitary point [p1,p2,p3]]

10. [Optional Challenge] Try to make sense out Attempted proof of RH by Nagy Adel, Mena Asham, and Marian Fady. (received by Dr. Z.'s on April 1, 2018).

Added April 9, 2018: Read the students' nice solutions   .

Programs done on April 5, 2018

Contains

• EvalCFg(A, B): evaluates a general continued fraction with numerators B and denominators A

• Tanh(n,x): the truncated continued fraction from Chrystal for tanh(x)

• FWB(n,x,p): the truncated continued fraction replacing 2*j-1 in tanh(x) by a general function so Tanh(n,x) is FWB(n,x,k->2*k-1).

• PadeExp(f,x,m,n): inputs a function f of the variable x and nonnegative integers m and n outputs a rational function of exp(x) with numerator of at most m and denominator of at most n such that the firs m+n-1 Taylor coefficients of f-MyPade(f,x,m,n) are zero

• [added after class] FWBg(n,x,y,p): Like FWB(n,x,p) by replacing x^2 by y

Homework for April 5, 2018 (due Sunday April 8, 2018, 10:00pm)

Subject: hw#21

and an attachment

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#YourName, Math 640 (Spring 2018) homework assignment 21

1. y(x)=tanh(x) happens to satisfy the differential equation y'(x)=y(x)^2-1

   Write a program

   GuessODE(f,x,N,k,y)

   that inputs a function f, or the the start of a Taylor series, in x, a positive integer N, a positive integer k, and a letter y, and outputs a polynomial in y, of degree k, such that f satisfies the ode

   f'(x)=P(f(x)),

   up to the N-th Taylor coefficient (and hopefully for ever). For example

   GuessODE(tanh(x),x,20,2,y) ;

   should return y^2-1 .

2. Using procedure

   FWBg(n,x,y,p)

   added in C21.txt after class (FWBg(n,x,x^2,j->2*j-1) gives tanh(x)), define the family of polynomials (depending on the integer parameters a and b)

   P_n(x,y) = numer(FWBg(n,x,y,j->a*j+b))   Q_n(x,y) = denom(FWBg(n,x,y,j->a*j+b))

   to form the sequence consisting of the first, say, 30, terms of each , and use procedure

   Findrec(Sequence,n,N,C)

   (with C=7) in the Maple package Findrec.txt to find the recurrences satisfied by these for various choices of a, and b. Can you conjecture a form for the recurrence(s) in terms of a and b?

Added April 9, 2018: Read the students' nice solutions   .

Programs done on April 9, 2018

C22.txt

Contains

• ZNt(N,t):zeta_N(1/2+I*t)*zeta_N(1/2-I*t), where Zeta_N(s) is the truncated Zeta function

• ZNtR(N,t): more efficient version, only using real-valued functions

• FindIC(f,t,T,res): inputs a non-negative function f of t, and a pos. nunber T and a resolution, and finds appx. minima

• OneS(f,t,t0): applies one step of the Newton-Raphson method

  RN(f,t,t0,N,err): estimates the closest zero of f of t to t0 as soon as two consecutive iterations are less than err apart, or we reached N iterations and return FAIL

Homework for April 9, 2018 (due Sunday April 15, 2018, 10:00pm)

Subject: hw#22

and an attachment

hw22FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 22

The rest of the homework, for this semester, will be working on the two class projects, where the templates are Truncated Zeta Project and Random Polygon Project. Soon we will divied the labor and get organized, but meanwhile please do all the problems below, that eventualy will be part of the class' papers.

1. For the truncated Zeta project, finish what we did in class and write a program

   TrunZetaMinima(N,T,err),

   that inputs a positive integer N, a large real number T, and a small error, (positive real number), and outputs the list of all pairs [location, value] where location is a local minimum of ZNtR(N,t), and value is its value there.

   Try to generated as much data as possible.
   [Added April 12, 2018: we did this one in C23.txt, so it is officially optional now (of course, you can make it better)]

2. For the random polygon project, get a feel for it, by
   • Writing a procedure

     RandomGoodPoly(n)

     that inputs a positive integer and outputs a random polygon with n vertices, whose centroid is the origin.

     Hint: generate two random vectors, x and y, and substract from them their averages, also call these new vectors x and y to make their sum to zero and let the polygon be [[x1,y1], ..., [xn,yn],[x1,y1]]

   • Write a procedure

     AvePolygon(P)

     that inputs a polygon, (given by its list of vertices), and outputs the polygon whose vertices are the midpoints of the edges.

   • Write a procedure

     IterPolygon(n,N)

     that inputs a RandomGoodPoly(n) and a large pos. integer N, and performs AvePolygon(P) N times, and also plots it (use the command plot(P)). Perform IterPolygon(20,200) many times and look at the picture.

3. [Challenge (no peeking in the internet)] Explain why this (limiting) pattern happens.

Added April 16, 2018: Read the students' nice solutions   .

Programs done on April 12, 2018

C23.txt

Contains

• Z(N,t): Zeta_N(1/2+I*t)*Zeta_N(1/2-I*t) where Z_N(s)=1+1/2^s+ ...+N^s, in floating point

• Ze(N,t): Zeta_N(1/2+I*t)*Zeta_N(1/2-I*t) where Z_N(s)=1+1/2^s+ ...+N^s Exactly (unevaluated)

• FindAZ(f,t,T,res): inputs a function f of t, a pos. integer T, and a pixel-size res and finds all the intervals [res*(i-1),res*i] where f changes sign

• OS(f,t,P): inputs a function f t and an interval P where the sign of f is opposite and outputs the left or right interval with the same property

• ZI(f,t,P,eps): inputs f,t,P, finds an approximation to the zero of f in P

• FindZ(f,t,T,res,eps): inputs a function f of t, a pos. integer T, and a pixel-size, res, and an error eps, and outputs (hopefully) all the approximate zeros in [0,T]

Homework for April 12, 2018 (due Sunday April 15, 2018, 10:00pm)

Subject: hw#23

and an attachment

hw23FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 23

1. Write a program

   GlobalMinimumZN(N,T)

   that inputs a positive integer N and a large positive integer T, and outputs the location and value of the global minimum of Z(N,t) in 0<=t<=T

   What are GlobalMinimumZN(N,1000) for N from 2 to 10?

2. Write a program

   GlobalMaximumZN(N,T)

   that inputs a positive integer N and a large positive integer T, and outputs the location and value of the global maximum of Z(N,t) in 0<=t<=T

   What are GlobalMaximumZN(N,1000) for N from 2 to 10?

Added April 16, 2018: Read the students' nice solutions   .

Programs done on April 16, 2018 (virtual class)

C24.txt

• IterPolygon(n,N): (written by Yukun Yao, taken from his hw22.txt) inputs a positive integer n, and a positive integer N, starts out with a random n-sided polygon with centroid at the origin, and keeps taking midpoints of adjacent vertices to form a new polygon, doing it N times.

• MtV(M,V): Multiplies a matrix M (given as a list of lists) and a compatible column-vector V (given as a list)

• LCdet(M): the determinant of the square-matrix M, using the method of Alice in Wonderland's creator (nicely proved here

• RandM(m,n,N): a randim m by n matrix (given as a list of lists) whose entries are between 1 and N

• ChM(M,x): M-x*I (as a list of lists)

• ChP(M,x):the characteristic matrix of the square matrix M, w.r.t the variable x, i.e. M-Ix

• EV(M,eps): the list of eigenvalues of the square matrix M, and decreasing order of their absolute values. eps is a small parameter, just for convenience

• EVC(M): inputs a square matrix M, given as a list of lists, and outputs the list of eigenvalues (in weakly-decreasing order of absolute value) and the respective UNIT eigenvectors, all in floating point.

• EVCm(M): does the same as EVC(M) BUT using Maple's built-in linalg package (I like it better than LinearAlgebra), but the eigenvalues are not sorted and the eigenvectors are not normalized to be unit

Homework for April 16, 2018 (due Wed. April 18, 2018, 10:00pm)

Subject: hw#24

and an attachment

hw24FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 24

1. Finish up EVCm(M) so that it will coincide with EVC(M), in other words, sort the list of eigenvalues into a weakly-decreasing list (by absolute value), and make all eigenvectors unit. Of course the eigenvectors should correspond to the respective eigenvalues.

2. Write a procedure

   Coeffs(V,M)

   that inputs a vector V (given as a list, of size n, say) and a list of vectors M (of the same size of V), and outputs the list a[1], ..., a[n] such that

   V=a[1]*M[1]+ ...+ a[n]*M[n]

   In other words express V as a linear combination of the vectors M[1], ..., M[n]

   [Hint: you can either use solve, or matrix inversion, using inverse(M) and the procedure MtV(M,V) created today]

3. Write a procedure, A(n), that inputs a positive integer n, and outputs the n by n matrix (given as a list of lists), corresponding to the "averaging process"

   [x1,x2, ..., xn] ->[(x1+x2)/2, (x2+x3)/2, ...., (xn+x1)/2]

Added April 18, 2018: Read the students' nice solutions   .

Programs done on April 19, 2018 (virtual class)

C25.txt

Contains procedures

• IterPolygon1(P,N): inputs a polygon P, performs the averaging N times and plots it.

• (From Edna Jones' hw24 )

• RandM(m,n,N): a random m by n matrix with entries from 1 to N (already present in C23.txt ):

• Nor(v): the normalized version of v (to have (L2) norm 1)

• Champs(L): given a list of pairs outputs the set indices of pairs with the largest first component according to absolute value

• EVC(M): the list of [eigenvalue,eigenvector] in weakly decreasing order of the absolute value of the eigenvalues

• CheckEVC(M,L): checks that EVC is (probably) correct

• [From Yonah B-A's hw24, adpated by Edna J.] Coeffs(V,M): inputs a vector V (of length n, say) and a list of (row vectors) M of length n, and outputs the vector A such that V=A[1]*M[1]+ ...+ A[n]*V[n]

• CB(M,V): inputs a matrix M and a vector V and outputs the list of coefficients expressing V in terms of the eigenvectors of M (arranged according to EVC)

Homework for April 19, 2018 (due Sun. April 22, 2018, 10:00pm)

Subject: hw#25

and an attachment

hw25FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 24

1. Prove that 1 is the an eignenvalue of A(n) and the all 1 vector is an eigenvector

2. [Added April 20,2018] plot the eigenvectors corresponding to the second-and-third largest (i.e. the largest ones that complex) eigenvetors of A(n) for n=30. Recall that its components are complex numbers and the complex number x+y*I corresponds to the point [x,y]. What kind of shape emerges? can you conjecture an equation for it?

3. [Possibly a challenge] Explain why the limiting shape is an elliplse
   [ Added April 21, 2018: previously I claimed that the inclination is 45 degrees, that was wrong. The inclination can be anything. The shape is an ellipse whose center is the origin, of course, but otherwise can be any shape and any orientation]

4. Write a program that inputs a good polygon and outputs the eccentricity [Added April 21, 2018: and orientation] of the limiting ellipe, by just using the initial points of the polygon (i.e. the vector of x-coordinates and the vector of y-coordiates, both of which add-up to 0).
   [Correction April 21, previously I said "add up to 1", of course that was wrong, by construction of a good polygon the x and y bectors add up to 0]

   Hints: (1) use Coeffs (2) Convince yourself that if A and B are any complex numbers, the curve
   x=Real(A*exp(I*t)), y=Real(B*exp(I*t))
   is an ellipse. By writing A=|A|*exp(I*arg(A)) and B=|B|*exp(I*arg(B)), can you find the eccentricity and orienation of that ellise?

5. [Challenge, 10 dollars to be divided; Added April 23, 2018: this prize was shared by Edna Jones and Ahsan Khan] Using experimentation, or otherwise, find an explicit expression, in terms of w:=exp(2*Pi*I/n), for the two largest eigenvalues of A(n) (they are complex-conjugates of each other) and if possible for the corresponding eigenvectors (they are also complext conjuates of each other).

Added April 23, 2018: Read the students' nice solutions   .

Programs done on April 23, 2018

• C25a.txt finishing up the iterated polygon. This was renamed IteratedPolygon.txt finishing up the iterated polygon, and will be hopefully expanded into an impressive Maple package for the Iterated Polygon class project.
• C26.txt , a Maple one-line proof of the moudular functional equation for the theta function, following George Polya's beautiful elementary proof, but letting Maple automatically do all the limits.

Homework for April 23, 2018 (due Sun. April 29, 2018, 10:00pm)

Subject: hw#26

and an attachment

hw26FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 24

1. Find a typo in Bellman's account of George Polya's beautiful elementary proof

2. [Challenge] figure out if and when the 45 degrees orientation holds, claimed by Van Loan in this chariming article .

3. [Challenge, I don't know the answer] Try to use Polya's brilliant idea of using an algebraic identity, l-secting it, plugging-in values and taking the limit, may yield other continuous, and who knows?, new, identitites. For example, start with the trinomial theorem (1/z+1+z)^(3*m).

4. Work as hard as you can on the class projects Truncated Zeta and Iterated polygon that you belong to (but feel free to contribute to the other one, if you will contribute significantly, the leader will add your name). [The leaders (Yukun Yao and Matt Charnley) should copy these pages, and the package to their own website, and at the very end, send them to me for posting]

Programs done on April 26, 2018

C27.txt

Contains, in addition to those of C23.txt, procedurs

• DioA(a,eps): the best (i.e. least denominator) appx. of a by a rational number with error
• LogRat(n,eps): The rational appx. of log(n) with error
• ZeRat(N,t,eps): the rational appx. to Ze(N,t) with error (of the log(n))

Homework for April 26, 2018 (due Tuesday, May 1)

Have the templates and the very preliminary versions in the respective leaders' websites the following fronts of the two projects
• Truncated Zeta Project
• Iterated Polygon project
Coordinate who will do what. The leaders will assemble all the pieces, and write a first draft of the papers and the Maple packages, that the team members can modify. It would be nice to have something postable by Aug. 31, 2018.

Programs done on April 30, 2018

C28.txt

Today we were getting ready for the field trip to the Princeton Cemetery (May 1, 2018) , by highlighting the work of some of the geniuses that we will visit.

Contains,

• VN(A,x,y): inputs a matrix A , m by n say, where player Row has m different strategies and player Column has n different strategies and the payoffs are A[m][n] and -A[m][n], respectively. Outputs the expected gain for Row (alias minus expected gain for Column) if they play a mixed strategy x and y

• VN2(A,x,y): the payoffs for 2 by 2 matrix and strategies [x,1-x], [y,1-y]

• RP(n): a random point in [0,1]^n

• RD(n): a random distance between 2 points in the n-dim unit cube
• MC(n,N): the average of the distances between two random points in [0,1]^n done N times

• RC3(): The value of the Robbins constant as a triple integral

• RC3e(): The excact value of the Robbins constant (due to David Robbins)

• RC2(): the exact value of the 2-dim Robbins constant

• RC44(): the exact value of the 4-dim Robbins constant

Optional Homework for April 30, 2018 (due whenever you feel like it)

[Please send it to DoronZeil@gmail.com rather than ShaloshBEkhad@gmail.com]

1. Implement, in Maple, the Cooley-Tukey. Compare its speed to the naive way of finding the Discrete Fourier Transform.

Added June 15, 2018: See the amazing paper by Yonah Biers-Ariel, Matthew Charnley, Brooke Logan, Anthony Zaleski and Dr. Z. that came out from one of the two group projects.

Added Jan. 29, 2019: See the amazing Maple package /TruncatedRiemannZeta.txt by Edna Jones and Yukun Yao. For beatiful pictures and data visit web-page of Truncated Riemann Zeta function project.