Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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5. Riemann's Zeta Function
5. Riemann's Zeta Function
Pages 63-81
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From page 63... ...
I mentione in Chapter l.iii that both Bernoullis found proofs for the divergence of the harmonic series. In the book where he published his brother's proof, and then his own, Jakob Bernoulli state(1 the above problem and asked anyone who could figure it out to tell him the answer.
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From page 64... ...
The mathematical term of art for this exact representation of a number is "closed form." A mere decimal approximation, however good, is an"open form." The number 1.6449340668... is an open form.
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From page 65... ...
The Basel problem opens the door to the zeta function, which is the mathematical object the Riemann Hypothesis is concerned with. Before we can pass through that door, though, I must recapitulate some essential math: powers, roots, and logs.
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From page 66... ...
I need to add a few more, because so far I have used only powers that are positive whole numbers. What about negative powers and fractional powers?
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From page 67... ...
Notice particularly the zero-th power of x, which is just a horizontal line at height 1 above the xaxis what mathematicians call"a constant function" (and Intensive Care Unit nurses call"a flat traced. For every argument x, the function value is 1.
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From page 68... ...
(X2, X3, X8) increase; and, much more to the point of this book, how slowly positive fractional powers like x0 5 (lo so.
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From page 69... ...
Division provi(les a complete solution to the problem of inverting multiplication. The analogy breaks down there because, while a x b is always, invariably and infallibly, equal to b x a, it is unfortunately not true, except occasionally and accidentally, that ab= ha.
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From page 70... ...
Figure 5-2 is a graph26 of log x, for arguments out to 55. I've particularly marked the function values for arguments 2,6,18, an(l 54.
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From page 71... ...
It means that, when face(1 with a (li~cult problem involving multiplication, by"taking logs" (i.e., by ap
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From page 72... ...
At first thought, that might seem to be very obvious. When I say"power of x," you probably think of squares and cubes; and you know that a graph of the squaring function or the cubing function zooms up out of sight as the argument increases, way beyond the feeble inching-up of the log function.
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From page 73... ...
If a is bigger than 1/ e, this is true already, even in this diagram. If a is less than 1 /e, then by going far enough east by taking a big enough argument x the log x curve eventually cuts the xa curve again, and then, forever after, lies below it.
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From page 74... ...
Consider this statement: "The function log x eventually increases more slowly than x° °° or x°°°°°l, or x00000001 , or...." Suppose I raise this whol/e statement to some power say, the hun(lre(lth power. (This is not a very rigorously mathematical procedure, I admit, but it gives a true result.)
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From page 75... ...
VI. As an illustration of what I said in section I about the search for closed-form solutions yielding important insights, Euler's solution of the Basel problem not only gave a closed form for the reciprocalsquares series; as a by-product, it also gave closed forms for 1+2~4 +31 +4~4 +5~4 +..., 1+2~6 +3~6 +4~6 +51 ...,andsoon.SolongasN is an even number, Euler's result tells you the precise value, as a closed form, of the infinite series shown in Expression 5-1.
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From page 76... ...
Precise values closed forms were known for all even numbers N while for the odd numbers, approximate values could be got by just adding up enough terms.
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From page 77... ...
He used"s" instead; and so momentous was that 1859 paper that every succeeding mathematician has followed him. In studies of the zeta function, the argument is always given as "s." Here then, at last, is the Riemann zeta function (zeta, written " ~ ," being the sixth letter of the Greek alphabet)
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From page 78... ...
That's a capital sigma, the eighteenth letter of the Greek alphabet, the Greek"s" (for "sump. The way it works is, you stick the pattern "under" (which actually means to the right, though we illogically say"under")
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From page 79... ...
This is pronounce(l"Zeta of s is defined to be the sum over all n of n to the power of minus s." Here,"all n" is understood to mean"all positive whole numbers n." IX. Having got the zeta function set up as a neat expression, let's turn our attention to that argument "s." We know, from Chapter l.iii, that when s is 1, the series diverges, so that the zeta function has no value.
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From page 80... ...
Well, then Expression 5-3 would look like this: 1 1 1 1 1 1 1 1 1 1 ¢(o) = 1+2O +3O +4O +5O +6O 7° 8° 9 10 11 By Power Rule 4, this sum is 1 + 1 + 1 + 1 + 1 + 1+..., which pretty obviously (liverges.
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From page 81... ...
That series diverges, so this one must, too. Sure enough, if you take the trouble to actually work out the sums and add them up, you see that the first ten terms add up to 5.020997899....
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