Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind
September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal
Saturday September 10, 2005
Learning Log Number 9

11:00 am Lethbridge, Alberta

Today is wet and cold, a fairly heavy rain and a temperature of +10 C. A quick check indicates we now have a heavy rainfall warning for all of southern Alberta. This will be terrible news for the farmers.

Review

I have begun reading "Prime Obsession" by John Derbyshire. I hope to stay with this book until the end and then return to Michael Spivak's "Calculus".


I read chapter 7 a few days ago, and chapter 8 a few minutes ago. Now to begin making notes for chapters 7 & 8 [p. 99 - 136].

On my last session, I had decided to start "doing" some mathematics, using Mathematica. This morning I am tentatively going to change my mind and continue reading the Derbyshire book. However I may change my mind again.

Before making today's notes, I have finally published this website to the university web server.

Content

Prime Obsession. Chapter 7 The Golden Key, and an Improved Prime Number Theorem [p. 99 - 117]

"The Golden Key is, in fact, just a way that Leonhard Euler found to express the sieve of Erastothenes in the language of analysis." [p. 100]

"... on the left-side ... we have an infinite sum running through all the positive whole numbers 1, 2, 3, 4, 5, 6, ... , while on the right-side we have an infinite product running through all the prime numbers 2, 3, 5, 7, 11, 13, ... ." [p. 106]

This equation is known as the Euler product formula, but Derbyshire's term "the Golden Key" may well become popular.

Some more background:

The function log x has a gradient (i.e. derivative) of 1/x.

The integral of is , which gives the area under the graph.

There is no algebraic function for the integral of . The usual symbol for this is Li(t) and it is called the log integral function. That is, Li(x) gives the area under the curve of beween 0 and x. "When calculating integrals, areas below the horizontal axis count as negative, so that the area to the right of 1 works to cancel out the area to the left, as t increases.

Li(N) is a much better estimate of p(N) than .

Prime Obsession. Chapter 8 Not Altogether Unworthy. [p. 118 - 136]

Chebyshev showed that If for some fixed number C, then C must be equal to 1.

He also showed that p(N)cannot differ from by more than about ten percent up or down.

Riemann's paper "On the Hypotheses that Lie at the Foundations of Geometry" provided the mathematical framework for Einstein's General Theory of Relativity.


Here are a few quotes from "Prime Obsession" that I enjoyed:
Quotation
Comment
"When jotting down the ideas that make up this book ... " [p. 107]

Lovely. I wonder what these notes look like? What were the ideas and how did he keep track of them?

"Integrals are connected to sums at a deep level. The integral sign - first used by Leibnitz in 1675 - is just an elongated 'S' for 'sum' " [p. 112] I knew this, but wanted it to be part of these notes.
"Riemann's doctoral thesis is a key work in the history of complex function theory. ... almost the first things you learn in a course on complex function theory are the Cauchy-Riemann equations for a function to be well behaved and worthy of further investigation. These equations first appear in their modern form in Riemann's doctoral dissertation. ... Riemann's doctoral thesis is, in short, a masterpiece." [p. 121]  
Riemann's habilitation thesis (in thanks for being admitted to the Academy) is titled, 'On the representability of a function by a trigonometric series' and "is a landmark paper, giving the world the Riemann integral, now taught as a fundamental concept in higher calculus courses." [p. 127]  
His lecture paper was titled 'On the Hypotheses that Lie at the Foundations of Geoemetry'. "This is one of the top 10 mathematical papers ever delivered anywhere, a sensational achievement." [p. 127]  
"I imagine that you are familiar with the idea contained in Einstein's General Theory of Relativity, that the three dimensions of space and one of time can be dealt with mathematically as a four-dimensional space-time, and that this four-dimensiional continuum is warped and puckered by the presence of mass and energy. From the Gaussian point of view, the geometry of this space-time would have been developed by imagining it imbedded in a five-dimensional continuum, in the way that Gauss though of his two-dimensional surfaces as embedded in ordinary three-space. That modern physicists do not think of space-time in this way is due to Riemann." [p. 128 - 129] Fascinating. I still have a Gaussian view. The Riemann view involves the concept of tensors (which I never studied, but which I think has something to do with n-dimensions).
"That great habilitation lecture is, in fact, as much a philosophical document as a philosophical one. ... What he was speaking about at its most fundamental was the nature of space." [p. 130]

A google search indicated that a copy of this paper is in a book by Michael Spivak called Differential Geometry, which is in the university library. I was surprised to not locate a copy of this paper on the web, given how important it is claimed to be.

   

Riemann's zeta function is a generalized infinite series that will turn out to have important consequences for studying the distribution of prime numbers. I also am not yet sure about the ideas underlying Dirichlet's contribution.


I want to see the Riemann paper and see if I can get a sense of it.

12:40 PM

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