Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
Currently Skimming:
7. The Golden Key, and an Improved Prime Number Theorem
7. The Golden Key, and an Improved Prime Number Theorem
Pages 99-117
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 99... ...
Chapters 1 and 5 were all about those infinite sums, leading up to a mathematical object named, by Riemann, "the zeta function"; Chapter 3 was concerned with primes, taking its lead from the title of Riemann's 1859 paper and proceeding from there to the Prime Number Theorem (PNT)
|
From page 100... ...
70 years or so after Euclid he developed his famous sieve method for finding prime numbers. It works like this.
|
From page 101... ...
49 61 77 91 7 · / · · ~ 23 37 65 79 107 95 109 11 25 13 41 53 . 55 67 83 97 101 · ~ 29 43 71 85 The first number left unscathed after 3 is 5.
|
From page 102... ...
=1+—+—+—+—+—+—+—+ ~ + ~ +~. J 3s 5s 7s gs 1ls l3s l5s 17S 1gs The subtraction eliminated all the even-numbere(1 terms from the infinite sum.
|
From page 103... ...
When doing the original sieve, I chose to leave each original prime standing, deleting only its multiples by 2, 3, 4, .... Here, I eliminate the original prime from the right-hand side in the subtraction, along with all its multiples.
|
From page 104... ...
I don't like fractions with fractional denominators any more than you do, and there is a useful bit of mathematical notation I can introduce here to save on typing.
|
From page 105... ...
Well, there is an equivalent thing for when I am mul/tipl/ying terms that all conform to a pattern, the n sign. That's a capital Greek letter "pi," for "product." Here is Expression 7-2 written using the n sign.
|
From page 106... ...
Expression 7-3 the Gol(len Key is actually name(l"the Euler product formula."36 It first saw the light of day, though arranged slightly differently, in a paper with the title variae observationes circa series infinitas, written by Leonhard Euler and published in 1737 by the St. Petersburg Academy.
|
From page 107... ...
At a later stage of the book's development, I thought I had better carry out authorial due diligence, so I went to a research library (in this case the excellent new Science, Industry and Business branch of the New York Public Library in midtown Manhattan) and pulled out the original paper from Euler's collected works.
|
From page 108... ...
At every point it has a definite numerical value, though, just as your automobile has a definite speed at any point while you are acceleratingnamely, the spee(1 you see if you glance at the speedometer. If you glance again an instant later, you see a slightly different speed; but at every point in time there is some definite speed.
|
From page 109... ...
The gradient of the curve at the point is the gradient of that unique touching straight line. The gradient of log x at the argument x= 10 turns out, if you measure it, to be ~0 .
|
From page 110... ...
-3X-4 -2X-3 _X-2 0 1 2X 3X2 ... Of course x° is just 1, its graph a flat horizontal line.
|
From page 111... ...
To calculate that area, you first figure out the integral function of X-4. That, by the general rule above, is - 3 x-3, that is, -l/3x3.
|
From page 112... ...
Mathematicians have a way to write this, X-4 ~X, read as "the integral of x to the minus fourth power, with respect to x, from 2 to 3." (Don't worry too much about that"with respect to x." Its purpose is to declare x as the main variable we are working with, whose integral has to be figured out. If there happen to be any other variables under the integral sign, they are just hanging out there; they are not being integrated.
|
From page 113... ...
Figure 7-4 shows a graph of it. I have changed my symbol for argument from x to t, because I have a use for x other than as a dummy variable.
|
From page 114... ...
At To every time we refer to the darn thing, we simply define it to be a new function, and issue it a certificate declaring it a sound and respectable function in good standing with its peers. This new function has the name "the log integral function." The usual symbol for it is Lit.
|
From page 115... ...
. This is not merely true; it is, in a manner of speaking, truer.
|
From page 116... ...
~ Dialog N Because the twiddle sign is transitive, the two things are equivalent, as can be seen in Figure 7-6.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.