8.6.05

HARMONIC DIVERGENCE. Stalking the Riemann Hypothesis, recently reviewed, offers an interesting starting point for the creation of the zeta function, namely, the reciprocals of the primes. Start with the harmonic series, 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... This series diverges. Intuitively, one can see this: 1/3 + 1/4 > 1/2; 1/5 + 1/6 + 1/7 + 1/8 > 1/2; the sum of the next 8 exceeds 1/2, ad infinitum; see this (.pdf) for details. On the other hand, 1 + 1/2 + 1/4 + 1/8 + ... converges to 2; but what about my red terms, 1 + 1/2 + 1/3 + 1/7 + ... That series diverges, but the linked file and this site will reveal that it is not easy to demonstrate it.

The book reveals a commonality between the Riemann hypothesis and some work in physics. I read on p. 177, "the numbers that describe the primal waves of the prime counting function are much like numbers that describe the energy levels of the fundamental wave functions of an 'average' heavy nucleus." Theoretical physicists and number theorists are working on related problems, forsooth.

There's also this instructive passage at p. 216. "Research in mathematics is frustrating. If it's not frustrating, you're probably tackling problems that are too easy ... your steady state is that you are stuck." But you persevere and you'll often make progress.

On to Book Review No. 15, which returns to my blue terms, the reciprocals of the squares? That yields 1 + 1/4 + 1/9 + ... which converges to the circular constant pi squared over 6, or pi2/6. Now we're into e: The Story of a Number, which is a history of a number of related problems, including squaring the conic sections (the rectangular hyperbola posed a number of problems, but Gregoire Saint-Vincent, using methods proposed by Pierre de Fermat, noted that the rectangles he used to approximate the area were all of equal size -- unit elasticity! There's also an excerpt from George Cheyene's 1734 Philosophical Principles of Religion that must have offered the typefetter some challenges. It's illustrated, and this description is priceless.
Let (as before) AC, AH be the Afymptotes of any Hyperbola DLF defined by this Equation yxn=1, in which the Abfciffa AK=x and Ordinate KL=y, and n is fuppofed either equal to or greater than Unity.
There's also an imagined conversation between Johann Bernoulli the elder and J. S. Bach on the use of the logarithmic spiral and the tempering of the musical scale. (The major scale has two half steps and five full steps in an octave ... 1 - 1 - 1/2 - 1 - 1 -1 - 1/2; the chromatic scale thus has twelve half steps in an octave, and the frequencies of each recurrence of a note an octave higher are twice as great. But if one attempts to provide all smaller intervals as integer multiples one runs into problems.)

And thus the connections among various subseries of the harmonic series. To obtain the natural base e, add 1 to 1 + 1/2 + 1/6 + 1/24 + ... and get 2.718281828... (there is a limiting value but it is not the solution of any polynomial); there is a connection between 1/x (integrating out the rectangular hyperbola) and the natural logarithms, and e raised to the power ix, where i is the square root of negative 1 and x is any number, equals cos(x) + i sin(x), two trigonometric functions.

What occurred to me upon reading this book was how badly my high school math teachers exposed me to the idea of radians and polar coordinates. We begin measuring angles in degrees, and these correspond to what we can do on a map. No skipper in my experience has ever ordered, "Helmsman, steer a course of pi over 6 relative to magnetic." Is there a more intuitive way to expose beginning students to the circular functions and the exponential by playing with the sub-series of the reciprocal series rather than all of a sudden dropping radians on students?

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