Prime Obsession: Notes

 

 

Chapter 1

Chapter 20: The Riemann Operator and Other Approaches

 

Chaos Theory development began with Poincare with his three body problem. However the real progress started in 1960s with the availability of fast computers. Ed Lorentz is considered a pioneer in the development of Chaos theory. In 1971 physicist Martin Gutzwiller found a way to to relate chaotic system on the classical scale with analogous systems down in the quantum world, by allowing the quantum factor

 

 

 

Alain Connes approach is instead of finding the matrix whose eigen values are zeroes of Riemann Zeta function. He actually constructed such matrix. This is no mean feat

 

 

  Favor

• Hardy’s 1914 result that infinitely many zeros lies on the critical line
• RH implies PNT to be true
• Coin Tossing argument.
• Landau and Bohr stating infinitely many zeros lie close to critical line.
• Algebraic result of Artin, Weil and Deligne.
  Not in Favor
• Riemann had no sound reason to support his statement
• Zeta function exhibits some very peculiar behavior up the critical line
• Littlewoods 1914 result

 

 

Epilogue: 

Riemann spent the last 4 years of his life in Italy. He died under a fig tree. His stone is not at its original place but can be found in a wall. His daughter lived up a long age.

Chapter 22: Either its true or else it isn’t

When John Derbyshire asked this question to Andrew Odlyzko. He answered this way. It is indeed a hard problem and no one wants to take chance. For example Hilbert thought RH would be solved in his life time. Fermat in his son’s life time and problem 7 would take forever. However the reversed happened. In this chapter he also talks about AIM institute set up by John Fry and Gerald Alexanderson.

Another anecdote he mentions of Keating that when he and his friend went to visit Gottingen for Riemann paper they had totally different requirement one was working on a Physics problem and the other on number theory and the librarian said the same paper will do as Riemann was working on both of these problems at the same time.

A lot of people believe that RH is true because of the sheer weight of evidence . But remember quote by Littlewood

“A long open conjecture in analysis generally turns out to be false. A long open conjecture in algebra generally turns out to be true.”

Alan Turing was a non-beliver in RH.

Author himself believes that it’s a long way before we can find the solution.

Odlyzko knows that it could be false because of S functions. One can decompose the zeta function into different parts, each of which tells you something different about zeta’s behavior. One of these parts is the so called S function. S hovers between -1 and 1. The largest value known is 3.2. There are strong reasons to think that if S were ever to get up to around 100, then the RH might be in trouble. A value near 100 is a necessary condition but not sufficient one. Selberg proved in 1946 that S is unbounded. But the rate of growth of S is so small and its beyond current computational ability.

Even though there are lots of theorem which depend on the truthfulness of RH but no one knows what kind of upheavel the proof will bring.

Chapter 15: Big Oh and Mobius Mobius function

This chapter introduces the Big Oh notation. I especially loved the story of Paul Turán and how he is remembered for uttering “Big Oh to one” at the time of his death. I finally understood what it meant.

Definition of Big Oh

Function A is big oh of function B, if for large enough arguments, the size of A never exceeds some fixed multiple of B

For example: A function which is Big Oh to 1 is always bounded by k times 1 where k is a fixed multiple and we see that functions like are not bounded by big oh to 1.

Von Koch’s result:  This result says that

Mobius function:

Motivation: 

Definition of Mobius Function

• Its domain is N, that is all natural numbers 1,2,3,4,5,….
• 
• if has a square factor
• if is prime, or the product of odd number of different primes.
• if is the product of an even number of different primes.

The consequence of this definition is we have

Mertens’s function: It is given by

Show that for first 10 numbers its given by 1,0,-1,-1,-2,-1,-2,-2,-2,-1

Why Merten’s function is important ?

The reason its important because if one can show that then it implies Riemann hypothesis is true. Infact a weaker version of this equality is sufficient to show for every number , no matter how small.

Chapter 14: in the grip of obsession
This chapter gives the story of what happened in uk. The two main characters of this chapter are Hardy and Little wood. They collaborated together. Besides writing a very popular text on number theory Hardy also wrote A mathematician’s apology. Hardy’ seminal contribution in proving number theory is his result.
Infinitely many of zeta function’s non trivial zeroes satisfy the Riemann Hypothesis that is they have real part 1/2.
Littlewood’s contribution is his 1914 result. Li(x)-pi(x) changes from positive to negative infinitely many times. Later skews put a bound on that and improved upon his on bound. However these numbers are very large. Though Littlewood and Hardy were both lifelong bachelors and son of teachers. Littlewood fathered at least two kids. He was good at sports.
We also meet Landau in this chapter. He wrote a very famous book on mnumber theory. He’d was exceptional work ethics. He wrote more than 250 paper and 7 books. Infact the book Campeche to be known as “handbuch” and was widely read and influential. He was born in to a wealthy family and was very proud of his home which he used to described as the best home in town which you cannot miss.

Chapter 11: Nine Zulu Queens Ruled China

This chapter is just an introduction to the number system. The author has come up with a mnemonic (Nine Zulu Queens Ruled China) and says that mathematicians think of numbers as Russian Dolls one inside another. The highlight for me in this chapter was finding the value of .  To find this begin with which means . The other example which had my attention is using complex numbers for our power series … Notice that it converges for all real numbers between . However if we plug the complex number it still converges. When we do that we get … The way to see that is to draw these complex numbers on the argand diagram. Starting with 1 then going up by then going left by then going down by and so on and it gives a visual representation that its converging towards .

Chapter 9: Domain Stretching

This chapter introduces and its corresponding power series .. While the series converges only when . The function converges for every other than . The moral of the story is that an infinite series might define a function only part of its domain.

+….. This does converge and its easy to show that it converges to

for all negative even integers.

Chapter 8: Not Altogether Unworthy

This chapter gives a glimpse of life during Riemann’s time. How he worked so hard. His habilitation was a major event that even Gauss was impressed. It took him over 5 years to finish his doctorate. He also mentioned Dedekind who also got his doctorate under Gauss. Then we are introduced to Pafnuty Chebyshev. The unusual name Pafnuty comes from a  Church father in the fourth century.

There are two results of Chebyshev. cannot differ from by more than ten percent up or down. This was an important paper because its use of step function might have inspired Riemann’s use of similar function in his 1859 paper. Chebyshev’s method were elementary. However Riemann used full power of complex function theory in his 1859 paper. These results were so striking that other mathematicians followed him, and the PNT was proved at last using Riemann’s non-elementary method and for a long time it was believed that without invoking complex function theory it not possible to prove PNT. However Selberg in 1949 proved PNT using elementary mathematics. More proofs using elementary math have been discovered but they are all difficult.

Chebysheve bias: There are more primes of the form than . The first time this bias is violated is at . The first zone of violation is the 11 primes from to

Later part it talks about Dedekind’s view of Riemann and how he fared as a professor. His association with Dirichlet. His wife Rebecca Dirichlet who was quite a socialite. The successive death of Gauss, Dirichlet and people in his own family. His 1859 paper was number of primes less than a given quantity was earth shattering.

 

Chapter 7: The Golden Key and the Improved Prime Number Theorem

This chapter starts off with Eratosthenese sieve method of finding prime. Then he derives the Euler Product Formula and he calls it Golden Key.

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He comments on his proof that its by Euler that he found the most accessible. Then he defines log function and tells us what is the meaning of Derivative and Integral. At last he closes the chapter with the introduction of and tells us that its a much better estimate of

Chapter 19: Turning the Golden Key

In this chapter he introduces the idea of function and mobius function.

…

We notice that some terms (the fourth, eighth, ninth) are missing here. This is actually the Mobius function

Chapter 6: The Great Fusion

This chapter is  about the fusion of arithmetic and analysis and created what we now call as analytic number theory.  The credit for this goes to German mathematician Dirichlet. Who was studying prime numbers. Dirichlet was a big fan of Gauss and always carried around his book. He also was revered by Riemann and was a great teacher. He would never make eye contact with his students. Either he was writing on the blackboard or if he was facing them sitting on his high lectern he would close his eyes and place his hands in front of his face and dictate. It also touches on the rise of German mathematicians. He showed that in 1800s there was only Gauss among the prominent mathematicians but in 1900s there were lot more and it was because of the policies (Fredrick and Humboldt reforms) and Gauss’s influence.

Chapter 5: Riemann’s Zeta Function

The chapter begins with the Basel series and its history and what mathematician’s mean by exact value. In the appendix he described that though its credited with Jakob Bernoulli to have brought this problem to attention to masses. It was originally proposed by Pietro Mengoli who was a professor at Bologna university. This chapter is the introduction to exponent rules and he defines and use this to prove the rules of logarithms. He describes log function and the usefulness of sigma notation. We also get introduced to Riemann Zeta function in this chapter. There are two different problems which should not be confused first the sum of the series . This sum is known for all even terms (i.e exact value ). The sum to odd power still elude us. The other problem is about the zeroes of this function i.e at what point this graph crosses x-axis. We know that it has infinite number of trivial zeroes at negative even integers like $\latex \{ …,-6,-4,-2 \} $ but there are also infinite number of them like . Riemann zeta function problem is about the zeroes of the function not the convergence of this series.

Chapter 4: On the shoulders of Giants

This chapter is mainly about the story of Gauss and Euler.

Gauss was born in a very poor family and went to a very poor high school. After the anecdote of solving the problem he was adopted by Carl Wilhem Ferdinand. We got to know a bit of history about Carl too. Especially his tragic death and how he succumbed to the wounds he got in war. After his death Gauss took a job in Gotingen and remained there for the rest of his life. Author talks about that Gauss was not the most socially adept person. He discovered many things but never published it as he had no desire to assert himself socially. But he also ruffled feathers of fellow mathematicians by claiming that he already had thought about those problems before and invariably he was speaking the truth. He wrote to Legendre that he discovered Least Square method before him. Apparently Legendre was the first person to have written about Prime Number Theorem (PNT). He described it as

Euler was a devout Christian. He lived in Russia and Berlin. He was student of Bernoulli and is known for many discoveries in Math. He was partially blind in his 30s and completely by 60s. He produced so much math that people were still struggling to publish even 70 years after his death. He also write popular math. “Letters to a German Princess” where he describes many questions like why sky is blue, why moon looks bigger when it rises e.t.c

Chapter 16: Climbing the critical line

This chapter begins with Hilbert’s 2nd important address in which he uttered his famous 6 words. “We must know, we shall know”, while he was facilitated at konisberge by given the key of the city. Then it talks about anti-semitism. How Gotingen lost many of her jewish professors. Especially the story of Landau is heart breaking as he couldn’t come to term with it (he was not allowed to carry on his lesson) and he later died in Berlin, many other flocked to US. At the time of writing of books they had calculated . The value of z to very large values. He defined it as

It also talks about Turing and his own fascination with Riemann’s hypothesis and how he tried to build a mechanical machine to compute zeros of Riemann’s zeta function. He committed suicide by eating an able laced in cyanide.

Chapter 17: A Little Algebra

This chapter talks about the basics of modern algebra and introduces its readers to the concept of Field and how the finite field could be infinite for ex. .  We are given information about the characteristic of a field is the number of 1s it take to add to zero. are all fields with characteristic 0. He introduces them to Emil Artin who in his Ph.D thesis opened up new approaches to Riemann Hypothesis. The idea is one can extend finite fields as basis for building other fields. So one can build fields that bear uncanny resemblance, in its broad properties, to Riemann’s zeta function.

In 1933 Helmut Hasse (Germany) was actually able to prove a result analogus to the Riemann Hypothesis for a certain category of those base fields. Which was extended to much wider class of objects by André Weil- these were “Weil Conjectures”. Which were proven by Pierre Deligne (Belgium) in 1973 and that fetched him a Field’s medal.

The last part is about Matrices and he gives an introduction to Matrices, their characteristic polynomial, eigen values and trace. He then brings in Hermetian matrices whose all diagonal entries are real and all other values reflected about diagonal are complex conjugates. However the eigen values are all real. Which is similar to Riemann Hypothesis. Thus we have a Hilbert-Polya Conjecture.

” The non-trivial zeros of the Riemann zeta function correspond to the eigen value of the Hermetial matrices”.

 

Chapter 3 The Prime Number Theorem

This chapter is about the observation which lead people to think about what is a prime number theorem. They saw that the if we take and consider values for it increases by 7. Which lead them to conclude that its a function.

He later explains the consequence of  prime number therorem.

The probability that is prime is . The $N-th prime number is

 

Chapter Golden Key

Why is Zeta function important ?

We know that harmonic series diverges. The next question that comes to mind is what about the series .

Thus  

What is Golden Key and how to derive it ?

Golden key is writing the above zeta function in the form of primes. One can derive it by multiplying by and subtracting it for all the infinite primes and then we are left with only

 

Chapter 10:  A Proof and Turning Point

This chapter talks about the proof of Prime number theorem. It was proven independently by both Hadamard (French) and Poussini (Belgium) mathematician. Even though Poussini proved it slightly earlier and Hadamard had seen his proof. Hadamard’s proof is more simple and elegant. The reason both of them proved almost at the same time (1996) was because a year before Van Mangoldt (German) had prove that one need not prove the full Riemann hypothesis to prove PNT (Prime number theorem). One can prove it by proving a weaker result “The non-trivial zeros of Riemann zeta function have real part less than 1”. Also it was almost a decade since Stieltjes had made a claim that he had the proof of Proof of a result stronger than Riemann Hypothesis.

The author also talks about Hadamard who lived up to 98, known also for 3 circle theorem in Complex Analysis, was strong supporter of Deryfus. Was married to same woman for 64 years and whose all three sons died before him. Later when his beloved grandson died it was too much for him. He also mentions his popular book “The Psychology of invention in the Mathematical Field”

Chapter 12: Hilbert’s Eighth Problem

This chapter talks about Hilbert. He classified most humanity as stupid and disowned his own son. Was popular lecturer and he made his name by giving existence proof of Gordon’s theorem (even though Gordon said it was not mathematics it was theology). He also gave clean proofs of and being transcendental. At the age of 38 addressed 2nd international conference of mathematician and came up with 23 problems. Riemann hypothesis was problem 8. Continnum hypothesis was problem 1. There are three anecdotes from his life (1. When he went to attend a funeral of his gifted student who gave an incorrect proof and at funeral speech started talking about math. 2. His tore pants attracted attention and Courant politely made aware of him and he said it had been torn for a while and nobody noticed that. 3. After a student quit his class to join Poetry he said he didn’t have enough imagination to be math).

Jorgen Gram was the first mathematician to find the first 15 non-trivial zeros and confirming each had a real part but no pattern in the imaginary part.