NOTES
CHAPTER 2
|
only two books in English that give anything like a comprehensive account of Riemann: Michael Monastyrsky’s Riemann, Topology and Physics (the 1998 translation by Roger Cooke, James King, and Victoria King) and Detlef Laugwitz’s Bernhard Riemann, 1826–1866 (the 1999 translation by Abe Shenitzer). Though they are mathematical biographies—that is, much more math than biography—both books give a good picture of Riemann and his times, with many valuable insights. |
4. |
I should think they were. The distance from Lüneburg to Quickborn is 38 miles as the crow flies—10 hours walking at a brisk pace. |
5. |
Hanover did not become a kingdom until 1814. Before that, its rulers were titled “Elector”—that is, they had the right to participate in electing the Holy Roman Emperor. The Holy Roman Empire was wound up in 1806. |
6. |
Ernest Augustus was the last but one king of Hanover. The kingdom was incorporated into the Prussian Empire in 1866, a key moment in the creation of modern Germany. |
7. |
Rankings vary, but he is almost always in the top three, usually with Newton and either Euler or Archimedes. |
8. |
Heinrich Weber and Richard Dedekind published that first edition in 1876. The most recent edition of the Collected Works was compiled by Raghavan Narasimhan and published in 1990. The German for “Collected Works” is Gesammelte Werke, by the way; and this is a phrase so often encountered in mathematical research that English-speaking mathematicians, in my experience, say it in German quite unselfconsciously. |
9. |
An Abelian function is a multivalued function obtained by inverting certain kinds of integrals. The term is hardly used nowadays. I shall mention multivalued functions in Chapter 3, complex function theory in Chapter 13, and the inverting of integrals in Chapter 21. |
CHAPTER 4
|
Though I have not described it properly in this book, the very observant reader will glimpse the Euler-Mascheroni number in Chapter 5. |
20. |
In the mathematics department of my English university, all undergraduates were expected to take a first-year course in German. Those like myself who had studied German in secondary school were shipped off to the nearby School of Slavonic and East European Studies to learn Russian, which our instructors considered to be the language of most importance to mathematicians, after German. There you have the legacy of Peter. |
21. |
I have taken this story from a hilarious account of Frederick’s relations with Voltaire written by the English wit and satirist Lytton Strachey in 1915 and found in his Books and Characters: French and English. |
22. |
Euler’s Latin is a stripped-down, racing version of the language, designed not to show off the writer’s superb grasp of Augustan style (which Euler probably could have done if he had wanted to—he knew the Aeneid by heart) but to communicate ideas as plainly as possible with a minimum of verbiage to readers much less concerned with form than with content. I shall give some actual examples in Chapter 7.v. |
23. |
The President of the Berlin Academy, Pierre Maupertuis, was accused by Swiss mathematician Samuel König, probably correctly, of having plagiarized Leibnitz’s work. Maupertuis called on the Academy to pronounce König a liar, which they duly did. Writes Strachey: “The members of the Academy were frightened; their pensions depended on the President’s good will; and even the illustrious Euler was not ashamed to take part in this absurd and disgraceful condemnation.” |
24. |
First English edition 1795; first American, 1833. For some reason this book can now be found only in expensive collector’s editions. |
CHAPTER 5
26. |
If the shape of the curve looks oddly familiar, that’s because if you add up N terms of the harmonic series (Chapter 1.iii), you get a number close to log N. In fact, and the profile of that tottering stack of cards, if you rotate it clockwise through 90 degrees then reflect it in a vertical mirror, is the graph of log x. |
27. |
Note: It is a convention in math to use ε —that’s epsilon, the fifth letter of the Greek alphabet—to mean “some very tiny number.” |
28. |
The proof was devised by Greek-French mathematician Roger Apéry, who was 61 years old at the time—so much for the notion that no mathematician ever does anything worthwhile after the age of 30. In honor of this achievement, the sum—its actual value is 1.2020569031 595942854…—is now known as “Apéry’s number.” It actually has some use in number theory. Take three positive whole numbers at random. What is the chance they have no proper factor in common? Answer: around 83 percent—to be precise, 0.83190737258070746868…, the reciprocal of Apéry’s number. |
CHAPTER 6
|
professional mathematicians. Then, around the middle of the twentieth century, there began to be a road back. Perhaps it all began with Harold Davenport’s 1952 book The Higher Arithmetic, an excellent popular presentation of serious number theory, whose title echoed an occasional synonym for “number theory” going back at least as far as the 1840s. Then, some time in the 1970s (I am working from personal impressions here) it began to be thought cute for number theorists to refer to their work as just “arithmetic.” Jean-Pierre Serre’s A Course in Arithmetic (1973) is a text for graduate students of number theory, covering such topics as modular forms, p-adic fields, Hecke operators, and, yes! the zeta function. I smile to think of some doting mother picking it out for her third-grader, to help him master long multiplication. |
31. |
The pronunciation of Dirichlet’s name gives a lot of trouble. Since he was German, the pronunciation should be “Dee-REECH-let,” with the hard German “ch.” English-speakers hardly ever say this. They either use the French pronunciation “Dee-REESH-lay,” or half-and-half it: “Dee-REECH-lay.” |
32. |
Constantin Carathéodory, though of Greek ancestry, was born, was educated, and died in Germany. Cantor was born in Russia and had a Russian mother, but he moved to Germany at age 11 and lived there practically all his life. Mittag-Leffler was the Swede. According to mathematical folklore, he was the cause of there being no Nobel Prize in mathematics. The story goes that he had an affair with Nobel’s wife, and Nobel found out. It’s a nice story, but Nobel was not married. |
33. |
Felix’s first cousin, Ottilie, married the great German mathematician Eduard Kummer; their grandson, Roland Percival Sprague, was co-creator of “Sprague-Grundy Theory,” in twentieth-century Game Theory…. I have to resist the temptation to take this further; it’s like tracing the genealogies of those German princes. Another Mendelssohn link will show up in Chapter 20.v. |
35. |
Mathematics allows infinite products, just as it allows infinite sums. As with infinite sums, some of them converge to a definite value, some diverge to infinity. This one converges when s is greater than 1. When s is 3, for example, it is The terms get closer and closer to 1 really fast, so at each step in the multiplication you are multiplying by something a teeny bit bigger than 1 … which, of course, hardly changes the result. Add 0 to something: no effect. Multiply something by 1: no effect. In an infinite sum, the terms have to get close to 0 really fast, so that adding them has very little effect; in an infinite product, they have to get close to 1 really fast, so that multiplying by them has very little effect. |
36. |
“Golden Key” is strictly my nomenclature. “Euler product formula” is standard. So are the following terms for the two parts, “the Dirichlet series” for the infinite sum, and “the Euler product” for the infinite product. Strictly speaking, the left-hand side is a Dirichlet series and the right-hand side is an Euler product. In the narrow context of this book, though, “the” is fine. |
37. |
There are two ways to define Li(x), both, unfortunately, in common use. In this book I shall use the “American” definition given in Abramowitz and Stegun’s classic Handbook of Mathematical Functions, published in 1964 by the National Bureau of Standards. This definition takes the integral from 0 to x, and this is also the sense in which Riemann used Li(x). Many mathematicians—including the great Landau (see Chapter 14.iv)—have preferred the “European” definition, which takes the integral from 2 to x, avoiding the nasty stuff at x = 1. The two definitions differ by 1.04516378011749278…. The Mathematica software package uses the American definition. |
38. |
You can get a good approximation for Li(N) by just adding up 1 / log 2, 1 / log 3, 1 / log 4, … , 1 / log N. If you do this for N equal to a million, for example, you get 78,627.2697299…, while Li(N) is equal to 78,627.5491594…. So the sum gives an approximation that is low by 0.0004 percent. That integral sign sure does look like an “S” for “sum.” |
CHAPTER 8
39. |
Mostly. Prussia and Austria also held parts of historic Poland. |
40. |
He worked for a year and a half as an assistant in Weber’s physics lab and might have earned some spare change thereby, so perhaps was not utterly without income. |
41. |
Topology is “rubber-sheet” geometry—the study of those properties of figures left unaffected by stretching, without tearing or cutting. The surface of a sphere is topologically equivalent to that of a cube, but not to that of a doughnut or a pretzel. The word “topology” was coined by Johann Listing in 1836, in a letter to his old schoolmaster. In 1847 Listing wrote a short book titled Preliminary Sketch of Topology. He was a professor of mathematical physics at Göttingen during Riemann’s time there, and Riemann certainly knew him and his work. However, Riemann seems never to have used the word “topology,” always referring to the topic by the Latin term favored by Gauss, analysis situs—“the analysis of position.” |
42. |
Eugene Onegin, 1833; A Hero of Our Times, 1840; Dead Souls, 1842. |
43. |
He was also the subject of a 1959 comic song, Lobachevsky, by mathematician/musician Tom Lehrer. |
44. |
Atle Selberg, now the Grand Old Man of number theory, is still at the Institute at the time of writing (June 2002) and still mathematically active. There is a story about this in Chapter 22. He was born June 14, 1917, in Langesund, Norway. |
45. |
Riemann, Gauss, Dirichlet, and Euler also enjoy this distinction. Riemann’s crater is at 87°E 39°N. |
46. |
I should perhaps explain that mathematicians have their own particular approach to the learning of foreign languages. To be able to read mathematical papers in a language not one’s own, it is by no means necessary to master that language thoroughly. You need to learn only the few dozen words, phrases, and constructions that are common in mathematical exposition: “it follows that…,” “it is sufficient to prove that…,” “without loss of generality…,” and so on. The rest is symbols like √ and Σ , that are common to all languages (though there are some minor national dialects in their usage). Some mathematicians, of |
|
course, are fine linguists. André Weil (see Chapter 17.iii) spoke and read English, German, Portuguese, Latin, Greek, and Sanskrit, besides his native French. I am speaking of ordinary mathematicians. |
47. |
Two of Gauss’s six children emigrated to the United States, where they helped populate the state of Missouri. |
CHAPTER 9
CHAPTER 10
50. |
Riemann’s Zeta Function, by H.M. Edwards (1974). Reprinted by Dover in 2001. |
51. |
A few unfortunate cases like Riemann notwithstanding, higher mathematics is wonderfully healthful. In writing this book, I have been struck by the number of mathematicians who lived to advanced ages, active to near the end. “Mathematics is very hard work, and dons tend to be above the average in health and vigor. Below a certain threshold a man cracks up, but above it hard mental work makes for health and vigor (also—on much historical evidence through the ages—for longevity).”—The Mathematician’s Art of Work by J.E. Littlewood, 1967. Littlewood, of whom I shall have much more to say in Chapter 14, was an illustration of his own argument. He lived to be 92. A colleague, H.A. Hollond, recorded the following note about him in 1972: “In his 87th year he is still working long hours at a stretch, writing papers for publication and helping mathematicians who send their problems to |
CHAPTER 11
57. |
Nowadays it is more often called “the argument” and denoted by Arg(z). I have used the older term, partly out of loyalty to G.H. Hardy (see Chapter 14.ii) and partly to avoid confusion with my use of “argument” to mean “the number to which a function is applied.” |
|
land) in 1800 had about 24 million people, the Hungarian-speaking population of Hungary was around 8.7 million in 1900, and I believe never rose above 10 million. This small and obscure nation produced an astonishing proportion of the world’s finest mathematicians: Bollobás, Erdélyi, Erdős, Fejér, Haar, Kerékjártó, two Kőnigs, Kürschák, Lakatos, Radó, Rényi, two Rieszes, Szász, Szegő, Szokefalvi-Nagy, Turán, von Neumann, and I have probably missed a few. There is a modest literature attempting to explain this phenomenon. Pólya himself thought that the major factor was Fejér (1880–1959), an inspiring teacher and gifted administrator, who attracted and encouraged mathematical talent. A high proportion of the great Hungarian mathematicians (including Fejér) were Jewish—or, like Pólya’s parents, “social” converts to Christianity, of originally Jewish stock. |
65. |
“The vertex figures of a regular polytope are all equal.” A polytope is the n-dimensional equivalent of a polygon in two dimensions, or a polyhedron in three. It is regular if all its “cells”—its (n – 1)-dimensional “faces”—are regular and all its vertex figures regular. The cells of a cube are squares; the vertex figures are equilateral triangles. Longevity watch: “Donald” Coxeter was born February 9, 1907. In late 2002, he was still listed as a faculty member of the University of Toronto. He published a paper, jointly with Branko Grunbaum, in 2001. Of the famously prolific Coxeter, a mathematician remarked to me: “Donald seems to have slowed down some recently.” |
66. |
Theory assures us, by the way, that the real part is precisely and mathematically , not 0.4999999, or 0.5000001. I shall say more about this in Chapter 16. |
CHAPTER 13
|
ematicians do.) Pólya used to tell his students that the common use of “z” for the argument and “w” for the value in complex function theory derived from the German words Zahl, which means “number,” and Wert, which means “value.” I don’t know if this is true, though. |
68. |
Estermann (1902–1991) made his mark in mathematics by proving, in 1929, that the Goldbach Conjecture, which asserts that every even number greater than 2 is the sum of two primes, is almost always true. He was also the originator of my proof for the irrationality of in Note 11—“the first new proof since Pythagoras,” he used to boast. |
69. |
Mathematicians working with functions of a complex variable generally say “the z plane” and “the w plane,” it being understood that “z” is the generic argument and “w” the generic value in complex function theory. |
70. |
And both kinds of illustration have really come into their own only with the advent of fast computer workstations and PCs. Before then, constructing pictures like my Figures 13-6 through 13-8 was an awfully painstaking business. |
CHAPTER 14
74. |
In the mid-1930s, the Soviet intelligence services recruited five young Cambridge undergraduates. Their names were Guy Burgess, Donald Maclean, Kim Philby, Anthony Blunt, and John Cairncross. This “Ring of Five,” as the Soviets referred to them, all went on to attain high positions in the British political and intelligence establishments during the 1940s and 1950s and passed vital information to the U.S.S.R. through World War II and the Cold War. Four of the five were at Trinity; Maclean was at Trinity Hall, a separate, smaller college. |
75. |
Lytton Strachey, Leonard Woolf, Clive Bell, Desmond MacCarthy, Saxon Sydney-Turner, and both Stephen brothers (Thoby and Adrian) were Trinity men. John Maynard Keynes, Roger Fry, and E.M. Forster, however, were at King’s. |
76. |
So it is always said. In his book on George Pólya, though, Jerry Alexanderson claims that the Pólya estate holds many more. |
77. |
Though the spine of my copy, a first edition, says simply “Primzahlen.” |
78. |
There are also lower bounds in problems of this sort. A lower bound is a number N for which we could prove that whatever the precise answer may be, it is certainly greater than N. In the case of the Littlewood violations, there seems to have been less work done here, presumably because everyone knew that the precise value of the first violation was extremely large. Deléglise and Rivat established 1018 as a lower bound in 1996 and have since extended the lower bound to 1020, but in view of the Bays and Hudson result, these lower bounds are almost nugatory. |
79. |
If the names Bays and Hudson ring a bell, that is because I mentioned them in Chapter 8.iv in connection with the Chebyshev bias. There is in fact a deep level, too deep to explore further here, at which the tendency of Li(x) to be greater than π (x) is kin to the Chebyshev biases. These two issues are generally dealt with as one by analytic number theorists. In fact, Littlewood’s 1914 paper showed not only that the tendency of Li(x) to be greater than π (x) is violated infinitely many times, but that this is also true of Chebyshev biases. For some very fascinating recent insights on this topic, see the paper “Chebyshev’s Bias,” by Michael Rubinstein and Peter Sarnak, in Experimental Mathematics, Vol.3, 1994 (pp. 173–197). |
80. |
Von Koch is better known to readers of pop-math books for the “Koch snowflake curve.” The “von” always gets dropped in that context, I don’t know why. |
CHAPTER 15
81. |
Either unaware of Bachmann’s book, or (more likely) just choosing not to use the new big oh notation, von Koch actually expressed his result in a more traditional form |
82. |
There has been a vast amount of research in this area. It is quite probably the case, in fact, that , which may be what Riemann meant by his “order of magnitude” remark. However, we are nowhere near being able to prove this. Some researchers, by the way, prefer the notation , to emphasize that the constant implied by the definition of big oh depends on ε . If you use this notation, the logic of Section 15.iii changes slightly. Note that the square root of N is about half as long (I mean, has about half as many digits) as N. It follows, though I shall not pause to prove it in detail, that Li–1(N) gives the N-th prime, correct to about half-way along, that is, roughly the first half of the digits are correct. The expression “Li–1(N)” here is to be understood in the inverse-function sense of Chapter 13.ix, with this meaning: “The number K for which Li(K) = N.” The billionth prime, for instance, is 22,801,763,489; Li–1(1,000,000,000) is 22,801,627,415— five digits, very nearly six, out of eleven. |
83. |
Möbius is best remembered for the Möbius strip, shown in Figure 15-4, which he discovered for himself in 1858. (It had previously been described by another mathematician, Johann Listing, also in 1858. Listing published, and Möbius didn’t, so according to the academic rules it should really be called “the Listing strip.” There is no justice in this world.) To create a Möbius strip, take a strip of paper, hold the two ends together (one in your right hand, one in your left), twist one end through 180°, and glue the ends together. You now have a one-sided |
CHAPTER 16
|
T2.” (See Figure 16-1.) Theory B tells you: “There are m zeros on the critical line from T1 to T2.” If it turns out that m = n, then you have verified the Riemann Hypothesis between T1 and T2. If, on the other hand, m is less than n, you have disproved the Riemann Hypothesis! (It is, of course, logically impossible for m to be greater than n.) Theory B deals with matters on the critical line. There is no possibility that the zeros being discussed here might have real parts 0.4999999999 or 0.5000000001. Compare the note on this in Chapter 12.vii. |
93. |
All the zeros computed so far appear to be irrational numbers, by the way. It would be astonishing and wonderful if an integer showed up among them, or even a repeating decimal (indicating a rational number). I know no reason this should not happen, but it hasn’t. |
94. |
The Fields Medal, first awarded 1936, was the idea of Canadian mathematician John Charles Fields (1863–1932). Now given at four-year intervals, its main purpose is to encourage promising younger mathematicians. Therefore, it is given only to those under 40. Several of the mathematicians named in this book have been Fields medalists: Atle Selberg (1950), Jean-Pierre Serre (1954), Pierre Deligne (1978), and Alain Connes (1982). The Fields Medal is held in high esteem by mathematicians. If you are a Fields winner, every mathematician knows it, and speaks your name with great respect. |
95. |
Not “104,” as Hodges says. |
96. |
The Theory of the Riemann Zeta-function (1951). Still in print. |
97. |
Just one more biographical note. Josef Backlund (1888–1949) is the other Finn in this book, born into a working-class family in Jakobstad on the Gulf of Bothnia. “The family was gifted but seems to have been mentally unstable; three brothers of Josef committed suicide.” (The History of Mathematics in Finland, 1828–1918, by Gustav Elfving; Helsinki, 1981.) A student of Lindelöf ’s, Backlund became an actuary after taking his doctorate and made a career in insurance, like Gram. Human knowledge owes a great deal to the insurance business. Gram, by the way, died an absurd death—struck and killed by a bicycle. |
98. |
Professor Edwards’s book includes some photographs of pages from the Nachlass, illustrating the scale of the task Siegel undertook. |
CHAPTER 17
CHAPTER 18
|
ing Sir Michael Berry. I have done my best to refer to him as “Berry” in writing of his activities up to 1996, and “Sir Michael” thereafter; but I don’t guarantee consistency. |
111. |
The Cray-1 was supplemented by a Cray X-MP at some point in the late 1980s. |
112. |
The earliest reference I have been able to track down to the Montgomery-Odlyzko Law thus named is in a paper by Nicholas Katz and Peter Sarnak published in 1999. The word “Law” is of course to be understood in a physical, rather than a mathematical sense. That is, it is a fact established by empirical evidence, like Kepler’s laws for the motions of the planets. It is not a mathematical principle, like the rule of signs. The Sarnak-Katz paper actually proved the law for zeta-like functions over finite fields (see Chapter17.iii), thus establishing a bridge between the algebraic and physical approaches to the RH. |
113. |
The answer is not “a half.” That would be to confuse the median with the average. The average of these four numbers: “1, 2, 3, 8510294,” is 2127575; but half of them are less than 3. |
114. |
Known to mathematicians as a “Poisson distribution.” The number e, by the way, is all over here. That 6,321, for example, is 10,000(1 – 1 / e). |
115. |
The equation I used for the curve in Figure 18-5 is y = (320000 / π 2) x2e–4x2/π . It is a skewed distribution, not (like the Gaussian-normal) a symmetrical one. Its peak is at argument , i.e., 0.8862269…. This was the curve surmised by Eugene Wigner for the GUE consecutive-spacings distribution. His surmise was based on the small amounts of data that can be gathered from experiments on the nucleus. It later turned out that this is not precisely the correct curve, though it is accurate to about a 1% error. The true curve, found by Michel Gaudin, has a more difficult equation. Andrew Odlyzko had to write a program to draw it. |
|
theory for the layman … unless you count Tom Stoppard’s 1993 play Arcadia. |
117. |
Hensel (1861–1941) was yet another branch of the Mendelssohn tree. His grandmother, Fanny, was the sister of the composer; his father, Sebastian Hensel, was her only son. Sebastian was 16 when Fanny died, and he was sent to live with the Dirichlets (Chapter 6.vii), with whom he remained until his marriage. Kurt spent most of his career as a professor at the University of Marburg, in central Germany, retiring in 1930. In spite of the Jewish lineage, he seems not to have suffered under the Nazis. “In general, the Mendelssohns did not feel the full brunt of the Nuremberg anti-Semitic laws because most of the family had undergone conversion several generations back.” (H. Kupferberg, The Mendelssohns.) In 1942, Hensel’s daughter-in-law donated his large mathematical library to the newly Nazified University of Strasbourg in occupied Alsace, reopened in November that year as the Reichs-universität Straßburg (but nowadays back in France once more). |
118. |
And at least one mathematician has expressed guarded skepticism in print. Reviewing Connes 1999 paper “Trace Formulae in Non-commutative Geometry and the Zeros of the Riemann Zeta Function,” Peter Sarnak (who is neither of my mathematicians X and Y) noted: “The analogies and calculations in the paper and its appendices are suggestive, pleasing and intricate and for these reasons this appears to offer more than just another equivalence of RH. Whether in fact these ideas and in particular the space X can be used to say anything new about the zeroes of L(s, λ) is not clear to this reviewer.” The L(s, λ) Sarnak refers to is one of those analogues of the Riemann zeta function I mentioned in 17.iii. |
119. |
The official name for this approach is “Denjoy’s Probabilistic Interpretation,” after the French analyst Arnaud Denjoy (1884–1974). Denjoy was Professor of Mathematics at the University of Paris, 1922–1955. |
120. |
“Touching the dull formulas with his wand, he turned them into poetry.”—Gunnar Blom, from the memorial essay included in Cramér’s collected works. Cramér (1893–1985) was yet another immortal. He died a few days after his 92nd birthday. |
121. |
I have borrowed this thought experiment from Chapter 3 of The Prime Numbers and Their Distribution, by Gérald Tenenbaum and Michel Mendès France (American Mathematical Society publications, 2000). |
122. |
A good article on this topic is “Is π Normal?” by Stan Wagon, in Mathematical Intelligencer, Vol. 7, No. 3. |
123. |
I have a preprint copy of a very recent paper by Hugh Montgomery and Kannan Soundararajan, titled “Beyond Pair Correlation,” and delivering another blow to the Cramér model. The last words in the paper are, “…it seems that there is something going on here that remains to be understood.” |
124. |
Mathematics and Plausible Reasoning (1954). |
125. |
Franklin has written a very good book about nonmathematical probability theory, The Science of Conjecture (2001). I reviewed this book for The New Criterion, June 2001. |
CHAPTER 21
|
a flat constant function—then it takes every value, with at most one exception. For ez, the exception is 0. |
128. |
Though the definition involves some ambiguities, on the resolution of which there is no general agreement. The Mathematica 4 software package, for example, provides Li(x) as one of its built-in functions—it calls it LogIntegral[x]. For real numbers, it is just as I described it—in fact, I used it to draw the graph of Li(x) in Chapter 7.viii. For complex numbers, however, Mathematica’s definition of the integral is slightly different from Riemann’s. Therefore, I didn’t use Mathematica’s LogIntegral[z] for these complex calculations. I actually set up in Mathematica as . |
129. |
Looking at this list with one eye and Figure 21-3 with the other, you can see that the tendency of the first few zeros to be sent to numbers with negative real parts is just a chance effect, and soon rights itself. |
130. |
In Figures 21-5 and 21-6, I have referred to the complex conjugate of the kth zero as the –kth zero. This is just a handy way of enumerating the zeros. It is, of course, not the case that . |
131. |
Note that 639 ÷ 1050 = 0.6085714…. For large numbers N, the probability that N is square-free is ~ 6 / π2, that is, 0.60792710…. Recalling Euler’s solution of the Basel problem in Chapter 5, you might notice that this probability is 1 / ζ(2). This is generally true. The chance that a positive whole number N chosen at random is not divisible by any nth power is indeed ~ 1 / ζ(n). Of all the numbers up to and including 1,000,000, for example, 982,954 are not divisible by any sixth power. 1 / ζ(6) is 0.98295259226458…. |
CHAPTER 22