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.