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Bernhard Riemann, age 32, makes some remarks while presenting a paper on the
prime number distribution at the Berlin academy.  One-third of the way into
the paper, he makes an observation that is now known as the Riemann conjecture
   [it's about the distribution of the zeros of the Riemann zeta-function; states
   that all non-trivial zeros of the Riemann zeta function  have real part 1/2.

   Riemann zeta function:  zeta (s) =  SUM 1/i^s, i from 1 to infty.
   This is only one of the forms (given on p.l
   diverges for s < 1.  For s = 1.1 is about 10, for s=2 its about 1.6 etc.

   It has zeros at all even negative numbers -2, -4, etc (trivial zeros).
   I don't understand this. The
   non-trivial zeros arise at complex numbers. and the claim is that all of
   these complex roots have real part = 1/2. ]

However, Riemann offers no proof:

  One would, of course, like to have a rigorous proof of this, but I have put
  aside the search for such a proof after some fleeting vain attempts because
  it is not necessary for the immediate objective of my investigation.
	- Riemann, August 1859 paper: "On the Number of Prime Numbers Less
	  Than a Given Quantity."

The problem lies un-noticed for decades, but then it gains momentum.  In
August 1900, David Hilbert, speaking at the Second International Congress of
Mathematicians at Paris:

	Essential progress in the theory of the distribution of prime numbers
	has lately been made by Hadamard, de la Vallée Poussin, von Mangoldt
	and others. For the complete solution, however, of the problems set us
	by Riemann’s paper “On the Number of Prime Numbers Less Than a Given
	Quantity,” it still remains to prove the correctness of an exceedingly
	important statement of Riemann...

The problem continues today, while other famous problems have been resolved:
	- Four-Color Theorem (originated 1852, proved in 1976)
	- Fermat’s Last Theorem (originated probably in 1637, proved in 1994)

Poorly positioned

Having said all this, I must say that this book is not the place to find out
about Riemann zeta function.  The problem is that Derbyshire writes as if he
wants little old churchladies and everyone else to understand the zeta
function zeros.  So he wants you to learn what a function is and how to define
logarithms, even introduce symbols like Sigma over a full page.
He is simply wasting his breath...  It takes him
80 pages to even reach a definition of the function, Unfortunately
churchladies have returned to their bibles and the maths readers are dizzy
because their glasses are fogged up by Derbyshire's wasted breath... in the
end, this is a book meant for nobody.

But there are still interesting tidbits here and there...

Excerpts

The sets dealing with numbers:
  N: Natural - 1,2,3,4...
  Z: Integers:  + zero + negative
  Q: Rational numbers - signed integers + ratios of integers
  R: Reals: + irrational numbers like sqrt(2)
  C: Complex numbers.
are mnemonicized as Nine Zulu Queens Rule China.  (ch.11)

Multiplying -1 x -1: Rule of Signs

  "What does it mean to multiply a negative by a negative"?  This is a major
  sticking point in arithmetic for many people.  The best explanation I have
  seen is by Martin Gardner: Consider a large auditorium filled with two
  kinds of people, good people, and bad people.  I define "addition" to mean
  "sending people to the auditorium".  I define "subtraction" to mean
  "calling people out of the auditorium."  I define "positive" to mean
  "good" (as in "good people", and "negative to mean "bad".  Adding positive
  numbers means sending some good people to the auditorium; Adding negative
  numbers means sending some bad people in, which decreases the net
  goodness.  Subtracting a positive number means calling out some good
  people; subtracting -ve num is calling out bad people (goodness goes up).
  Thus, adding a -ve num is just like subtracting a positive.
  Multiplication is like repeated addition.  Minus three times minus five?
  Call out five bad people.  Do this three times.  Result?  Net goodness
  increases by 15.  When I tried this out on 6-yeear-old Daniel
  Derbyshire, he said, "What if you call for the bad people to come out and
  they won't come?  A moral philosopher in the making!"
	- p. 367-368, long footnote tangential to the discussion on p.42

Blurb:
In August 1859 Bernhard Riemann, a little-known 32-year old mathematician,
presented a paper to the Berlin Academy titled: "On the Number of Prime
Numbers Less Than a Given Quantity." In the middle of that paper, Riemann
made an incidental remark a guess, a hypothesis. What he tossed out to the
assembled mathematicians that day became the Riemann Hypothesis
Its resolution seems to hang tantalizingly just beyond our grasp.... , holds
the key to a variety of scientific and mathematical investigations. The making
and breaking of modern codes, which depend on the properties of the prime
numbers, have roots in the Hypothesis. In a series of extraordinary
developments during the 1970s, it emerged that even the physics of the atomic
nucleus is connected in ways not yet fully understood to this strange
conundrum.