Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

February 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

7:00 am Ballina NSW Australia

This is a continuation of my notes for "Dr. Riemann's Zeros".

Chapter 2 'Gorgeous stuff'

• The chapter begins with a discussion of continuing fractions. Let

This may be rewritten as , since the continuing expression is actually the same as the original expression.

From here it is easy to solve for a, obtaining , which is also known as the Golden Mean.

• "... the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary, they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification." [p. 32]
  
  
• "To summarize so far:
  What is the total number of primes less than any number n?
  Gauss's guess: n/log n - out by several per cent.
  Riemann's first guess: RF(n) - out by a fraction of 1 per cent.
  Riemann's better guess: RF(n) minus the sum of an infinite series S - perfect" [p. 33]
  
  
• S, the sum of the infinite series , is called the zeta function ( z ).
  
  
• "Functions are everywhere in mathematics, and they all work by putting some number in one end and seeing what comes out the other. But it's the relationship between the succession of inputs and outputs, rather than individual pairings of numbers, which the mathematicians are really interested in." [p. 34]
  
  
• "The mathematics of functions is called analysis, and one branch of analysis is very useful for looking at whole numbers. It's called analytic number theory." [p. 35]
  
  
• "Euler made what one mathematician has described as 'one of the most remarkable discoveries in mathematics'. He discovered that the zeta function can be expressed as a series of terms in which adding all the whole numbers was exactly equivalent to another function which consists of multiplying together a series of terms involving all the prime numbers. So feeding the same value for s into each of these two very different functions produced the same result." [p. 35]
  
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