Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)

A fact I learned at school in England by means of the following Victorian ditty:

George the First was always reckoned

Vile; but viler George the Second.

No one ever said or heard

A decent thing of George the Third.

When to heaven the Fourth ascended,

God be praised!—the Georges ended.

In fact, they did not end; the twentieth century brought forth two more Georges.

There was another great Elbe flood in 1962, causing many deaths and much destruction in the Wendland district. Following that, a system of major dikes was built. In August 2002, as I was finishing this book, the Elbe flooded again. However, the post-1962 dikes appear to have held, and the region has suffered less than those further upriver.

Erwin Neuenschwander is professor of the history of mathematics at the University of Zürich. He is the leading authority on the life and work of Bernhard Riemann and has edited Riemann’s letters. I have made use of his researches in this book. I have also relied heavily on the

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.

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.

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.

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.

Rankings vary, but he is almost always in the top three, usually with Newton and either Euler or Archimedes.

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.

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.

Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add it to the first. Keep doing this, piling on random numbers. How many random numbers,

on average, do you need to make the total greater than 1? Answer: 2.71828….

One of the great mathematical discoveries of antiquity, made by Pythagoras or one of his followers around 600 B.C.E., was that not every number is either a whole number or a fraction. The square root of 2, for example, is obviously not a whole number. Brute arithmetic shows that it is between 1.4 (whose square is 1.96) and 1.5 (whose square is 2.25). It’s not a fraction either, though. Here is a proof. Let S be the set of all positive whole numbers n for which the following thing is true: is also a positive whole number. If S is not empty, it has a least member. (Any non-empty set of positive whole numbers has a least member.) Call this least member k. Now form the number . It is easy to show that (i) u is less than k, (ii) u is a positive whole number, (iii) is also a positive whole number, so that (iv) u is a member of S. This is a contradiction, since k was defined to be the least member of S, and therefore the founding assumption— that S is not empty—must be false. Therefore, S is empty. Therefore, there is no positive whole number n for which is a positive whole number. Therefore, is not a fraction. A number that is neither whole nor fractional is called “irrational,” because it is not the ratio of any two whole numbers.

Rule of signs: a minus times a minus is a plus. This is a major sticking point in arithmetic for a lot of people. “What does it mean to multiply a negative by a negative?” they ask. The best explanation I have seen is one of Martin Gardner’s, as follows. Consider a large auditorium filled with two kinds of people, good people, and bad people. I define “addition” to mean “sending people into the auditorium.” I define “subtraction” to mean “calling people out of the auditorium.” I define “positive” to mean “good” (as in “good people”) and “negative” to mean “bad.” Adding a positive number means sending some good people into the auditorium, which obviously increases the net quantity of goodness in there. Adding a negative number means sending some bad people in, which decreases the net goodness. Subtracting a positive number means calling out some good people—net goodness in the auditorium

decreases. Subtracting a negative number means calling out some bad people—net goodness increases. Thus, adding a negative number is just like subtracting a positive, while subtracting a negative is like adding a positive. Multiplication is just repeated addition. Minus three times minus five? Call out five bad people. Do this three times. Result? Net goodness increases by 15…. (When I tried this out on 6-year-old Daniel Derbyshire, he said, “What if you call for the bad people to come out and they won’t come?” A moral philosopher in the making.)

One reader of this book’s manuscript thought that “twiddle” sounds like a Britishism. (I was educated in England.) I agree, it does. American mathematicians certainly use it, though. I have heard, for example, Nicholas Katz of Princeton University use it in a lecture. Prof. Katz is from Baltimore and was educated entirely in the U.S.A.

George was the last king of Hanover. The kingdom was swallowed by Prussia in 1866, after taking the wrong side in the Austro-Prussian war of that year. The medal seems not to have actually been struck until the Gauss centenary in 1877.

Among the Duke’s claims to fame, perhaps it is worth noting that he was the father of Caroline of Brunswick, who was married off to the Prince Regent of England. The marriage was a disaster and Caroline left England; but when the Prince ascended the English throne as King George IV, she returned to claim her rights as Queen. This caused a constitutional crisis of the minor sort, as well as much public merriment over the unpopular king’s discomfiture, his queen’s rather bumptious personality, her peculiar personal habits, and her flagrant liaisons. The following ditty was widely circulated.

Gracious Queen, we thee implore

To go away and sin no more;

But if this effort be too great,

To go away, at any rate.

One of the Duke’s maternal aunts married a Holy Roman Emperor and begat Maria Theresa, the great Hapsburg empress. Another mar-

ried Alexis Romanov and was the mother of Peter II, nominal Tsar when Leonhard Euler came ashore in St. Petersburg (Section VI of this chapter). Once you start following the genealogies of these petty German rulers, there is no end to it.

Did I mention that as well as being a towering mathematical genius and a physicist of the first rank, Gauss was also a brilliant astronomer, the first person to correctly compute the orbit of an asteroid?

To find out if some number N is prime, you just keep dividing it by primes 2, 3, 5, 7, … one after another until either one of them divides exactly, in which case you have shown that N is not prime, or … what? How do you know when to stop? Answer: you stop when the prime you are about to divide by is bigger than . Suppose N is 47, for example; is 6.85565…, so I only need to try division by 2, 3, and 5. If none of them works, 47 must be prime. Why don’t I need to try 7? Because 7 × 7 = 49, so if 7 divided exactly into 47, the quotient would be some number less than 7. Likewise, is 837.2574…. The last prime below this is 829; the next prime above it is 839. If 839 divided into 701,000, the quotient would be a number less than 839; either some prime less than 839 (which I would therefore already have tried), or a composite number made up of even smaller prime factors….

Legendre died in poverty, having offended his political superiors by taking a stand on principle. His dates are 1752–1833. I am sorry that I have presented him here as a disgruntled and slightly comical figure. Legendre was a fine mathematician, at the top of the second division, and did valuable work over many years. His Elements of Geometry was the leading elementary textbook on the subject for over a century. It is said to have inspired the tragic Évariste Galois—the narrator in Tom Petsinis’s novel The French Mathematician—to take up a career in mathematics. More relevant to the present narrative, his book Theory of Numbers—the renamed third edition of the Essay mentioned in the text—was lent by a schoolmaster to the adolescent Bernhard Riemann, who returned it in less than a week with the comment, “This is truly a wonderful book; I know it by heart.” The book has 900 pages.

There is a very good account of the Euler-Mascheroni number in Chapter 9 of The Book of Numbers, by John Conway and Richard Guy.

Though I have not described it properly in this book, the very observant reader will glimpse the Euler-Mascheroni number in Chapter 5.

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.

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.

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.

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.”

First English edition 1795; first American, 1833. For some reason this book can now be found only in expensive collector’s editions.

It had been posed by Pietro Mengoli in 1644. Mengoli was a professor at the University of Bologna at the time, so we really should say “the Bologna problem.” It was Jakob Bernoulli who first brought the problem to the attention of a wide audience, though, and “the Basel problem” has stuck.

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.

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.”

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.

English edition published by Bloomsbury USA, 2000. The novel was first published in Greek in 1992. As Doxiadis points out, the conjecture was first framed in proper mathematical form by Euler.

Of topics like the Goldbach Conjecture and Fermat’s Last Theorem, you might want to say “Oh, that’s not arithmetic, that’s number theory.” These two terms have had an interesting relationship. The phrase “number theory,” or at any rate “theory of numbers,” goes back to at least Pascal (1654, in a letter to Fermat), but was not clearly distinct from “arithmetic” until the nineteenth century. Gauss’s great classic on number theory was titled Disquisitiones Arithmeticae (1801). It seems to have been sometime in the later nineteenth century that “arithmetic” was definitely reserved for the basic manipulations learned in elementary school, with “number theory” used for the deeper researches of

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.

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.”

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.

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.

“Eratosthenes” is pronounced—at any rate by mathematicians—“era-TOSS-the-niece.”

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.

“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.

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.

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.”

Mostly. Prussia and Austria also held parts of historic Poland.

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.

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.”

Eugene Onegin, 1833; A Hero of Our Times, 1840; Dead Souls, 1842.

He was also the subject of a 1959 comic song, Lobachevsky, by mathematician/musician Tom Lehrer.

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.

Riemann, Gauss, Dirichlet, and Euler also enjoy this distinction. Riemann’s crater is at 87°E 39°N.

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.

Two of Gauss’s six children emigrated to the United States, where they helped populate the state of Missouri.

“Heck of a formula….” It is not actually so daunting, unless you have forgotten all your high school math. Other than the zeta function, there is nothing in there that high school math doesn’t cover, at least in part. The sine and factorial functions are, as mathematicians say, “elementary,” so this formula “elementarily” relates the value of zeta at argument 1 – s to its value at s. This formula, by the way, is called “the functional equation.”

A fact first proved by Bernhard Riemann, incidentally.

Riemann’s Zeta Function, by H.M. Edwards (1974). Reprinted by Dover in 2001.

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

him.”—Quoted by J.C. Burkill in Mathematics: People, Problems, Results (Brigham Young University, 1984).

I cannot restrain myself. “If f is an analytic function in the annulus , r is some number between r₁ and r₂ exclusive, and M₁, M₂, and M are the maxima of f on the three circles corresponding to r₁, r₂, and r, respectively, then .”

Stieltjes’s dates are 1856–1894. The most popular pronunciation of his name among English-speaking mathematicians is “STEEL-ches.”

“Reports Received.” This term is so common in scholarly bibliographies, it is often abbreviated to “C.R.”

He did not join the Communist Party, though his daughter Jacqueline did.

Though the glory of proving the PNT belongs to Hadamard and de la Vallée Poussin equally, I have written a great deal about the former and next to nothing about the latter. This is only in part because I find Hadamard an interesting and sympathetic character. It is also because there is much less material on de la Vallée Poussin. Though a fine mathematician, he appears to have been active in no other sphere. I mentioned this to Atle Selberg, the only mathematician I have spoken with who might have known both men. Hadamard? “Oh, yes. I met him at the Cambridge Congress” (i.e., in 1950). De la Vallée Poussin? “No. I never met him, and I don’t know anyone who did. I think he did not travel much.”

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.”

I do not mean to dismiss Kronecker as a crackpot. The case he made was, though I disagree with it, subtle and mathematically sophisticated.

For a spirited defense of Kronecker, see Harold Edwards’ article in the Mathematical Intelligencer, Vol. 9, No. 1. Kronecker was, says Prof. Edwards, “reasonable, not vitriolic.”

In German, Wer von uns würde nicht gern den Schleier lüften, unter dem die Zukunft verborgen liegt, um einen Blick zu werfen auf die bevorstehenden Fortschritte unserer Wissenschaft und in die Geheimnisse ihrer Entwickelung während der künftigen Jahrhunderte?

Hilbert actually only presented 10 of the problems to his audience, having been urged by those who had read the printed form of his address to shorten it for delivery. All 23 problems are listed in the printed address, and they are generally referred to by their numbers in that paper. The ones he actually read out to his audience at the Sorbonne were numbers 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22. A further confusion arises from the fact that some of Hilbert’s 23 bullet points just single out areas for investigation, and are only arguably problems. Typical is number 2, “To investigate the consistency of the axioms of arithmetic.” This accounts for the different numbering schemes you will sometimes see. Andrew Hodges, for example, in his biography of Alan Turing, counts 17 Hilbert problems, not 23, with the proof of the Riemann Hypothesis at number 4, not 8. Those of Hilbert’s items that were actual well-defined problems have now all been solved, with the single exception of the Riemann Hypothesis.

The best such book-length account that I know of is Jeremy J. Gray’s The Hilbert Challenge (Oxford University Press, 2000).

For a good popular account, see John L. Casti’s book Mathematical Mountaintops (Oxford University Press, 2001).

Most mathematicians of the time would have given that title to Henri Poincaré (1854–1912). The Hungarian Academy of Sciences in fact did so in 1905, awarding Poincaré its first Bolyai Prize as “that mathematician whose achievement during the past 25 years have most greatly contributed to the progress of mathematics.” The second Bolyai Prize was awarded to Hilbert in 1910.

George Pólya (1887–1985). Look at those dates—another immortal. Pólya was Hungarian. Even more striking than the rise of the Germans in the early nineteenth century was the rise of Hungarians in the early twentieth. While the German states (excluding Austria and Switzer

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.

“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.”

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.

Note incidentally, that the “unknown” complex number is most commonly represented by “z,” not “x.” Mathematicians customarily use “n” and “m” for whole numbers, “x” and “y” for real numbers, and “z” and “w” for complex numbers. We can, of course, use any other letters we feel like using—this is just a custom. (For the argument of the zeta function, I shall persist in that other custom of calling it “s,” as all math-

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.

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.

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.

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.

E.W. Barnes, Littlewood’s director of studies. He later became an Anglican bishop.

Author of Calcul des Résidus, a textbook of complex function theory. Ernst Lindelöf (1870–1946) was a great hero of Scandinavian mathematics, which he worked hard to advance through teaching, research, and writing textbooks. Born in Helsinki, he began his life a subject of the Russian Tsar—Finland did not get independence from Russia until 1917. Lindelöf was, however, a Finnish patriot (one of only two Finns in this book), and participated enthusiastically in the life of the new nation. He was the originator of the Lindelöf Hypothesis, a famous conjecture about the Riemann zeta function, concerning its rate of growth in the critical strip. I describe this conjecture in the Appendix.

A fellowship at Trinity was a lecturing position, with a regular stipend, and the right to take rooms in the college and eat dinner in the “hall” (refectory). It was not necessarily tenured.

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.

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.

So it is always said. In his book on George Pólya, though, Jerry Alexanderson claims that the Pólya estate holds many more.

Though the spine of my copy, a first edition, says simply “Primzahlen.”

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.

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).

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.

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

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.

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

strip—an ant can walk from any point on the strip to any other point without going over the edge.

FIGURE 15-4 A Möbius strip, with ant.

In case you think it was somewhat vainglorious of Möbius to pick a symbol equivalent to his own initial, let it be known that Möbius himself did not use μ when he first described the function in 1832; the μ is due to Franz Mertens in 1874, and Mertens was honoring Möbius, by then dead, not himself.

If the logic there escapes you, here’s a parallel case. Imagine that Theorem 15-1 said “All human beings are less than 10 feet tall,” while the Riemann Hypothesis said “All U.S. citizens are less than 10 feet tall.” If the first is true, the second must be true, since every U.S. citizen is a human being. The weaker result follows from the stronger one. If a human being 11 feet tall were discovered living in the remote highlands of New Guinea, then the existence of that person would prove Theorem 15-1 to be false. The Riemann Hypothesis, however, would still be open, since the giant is not a U.S. citizen. (Though I suspect he soon would be….)

Bernstein became a professor only in 1921. I have seen it written that he was also technically exempt under the Hindenburg modifications, but I do not know the basis for this statement. Bernstein (1874–1956) fled to the U.S. during the Hitler period but returned to Göttingen in 1948.

Carl Siegel told Harold Davenport the following story. “In 1954, to celebrate the 1,000th anniversary of Göttingen’s founding, the city fathers decided to give the freedom of the city to three of those professors who had been dismissed in 1933. The Tageblatt sent a reporter to Rellich (i.e., Franz Rellich, then director of the university’s Mathematics Institute) to ask if he could write an article on the three. Rellich replied, ‘Why don’t you just look up what you wrote back in ’33?’”

There is actually a branch of geometric function theory known, not altogether accurately, as “Teichmüller Theory.” It deals with the properties of Riemann surfaces. Teichmüller volunteered for active duty in World War II. He disappeared in fighting along the Dnieper in September 1943.

In the world of mathematics another instance was Ludwig Bieberbach, author of a famous conjecture in complex function theory (proved in 1984 by Louis de Branges). In 1933 at Berlin University, Bieberbach was conducting spoken examinations of doctoral candidates in full Nazi uniform.

I can think of no satisfactory English translation for Nachlass. Neither, to judge from the word’s frequent appearance in English-language materials, can anyone else. “Literary remains,” says my German dictionary. In this context the meaning is “unpublished papers found among a scholar’s effects after his death.”

Recall from my explanation of big oh that it involves some fixed constant multiplier. Thus, O(log T) means “This term never exceeds some fixed multiple of log T.” To describe the formula as “very good” is to say that the fixed multiplier is small. In this case it is less than 0.14.

This particular piece of theory deals with zeros actually, precisely, mathematically on the critical line. It is important to grasp the logic here. Theory A tells you: “There are n zeros in the rectangle from T₁ to

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.

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.

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.

Not “104,” as Hodges says.

The Theory of the Riemann Zeta-function (1951). Still in print.

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.

Professor Edwards’s book includes some photographs of pages from the Nachlass, illustrating the scale of the task Siegel undertook.

For example, S.J. Patterson, in his book, An Introduction to the Theory of the Riemann Zeta-Function, §5.11, wrote: “The most convincing reason that has so far been evinced for the validity of the Riemann Hypothesis is that an analogous statement is valid for the zeta-functions attached to curves over finite fields. The formal similarities are so striking that it is difficult to believe that they do not lead to even more far-reaching coincidences.” (My italics.)

To coin an apothegm, algebraists do not care so much about what things are, as about what you can do with them. They are verb people, not noun people. Another interesting conceptual perspective on algebra was offered by Sir Michael Atiyah at a Fields Lecture in Toronto in June 2000. While geometry is obviously about space (said Sir Michael, a Fields Medal winner), algebra is about time. “[G]eometry is essentially static. I can just sit here and see, and nothing may change, but I can still see. Algebra, however, is concerned with time, because you have operations that are performed sequentially….” (Shenitzer, A. and M.F. Atiyah. “Mathematics in the 20th century,” American Mathematical Monthly, Vol. 108, No. 7.)

Pronounced “Vay” by most English-speaking mathematicians. The main thing is to avoid listeners’ confusing him with Hermann Weyl (“Vail”). Weil, one of the most illustrious names in twentieth-century mathematics, was the brother of the mystic and French Resistance heroine Simone Weil. He had been a student of Hadamard’s at the Collège de France.

It might be better to say “from 1 to N zeros,” because zeros sometimes repeat. The zeros of the polynomial x² – 6x + 9 are 3 and 3. It factorizes as (x – 3)(x – 3). You might, therefore, prefer to say that this polynomial has only one zero, namely 3. In strict mathematical terms, this is “a zero of order 2.” There is a way to assign a similar order to any zero of any function, by the way. So far as we know, all the non-trivial zeros of the zeta function have order 1; but this has not been proved. Should a non-trivial zero of the zeta function show up with order 2 or greater, it would not disprove the Hypothesis, but it would create havoc with some of the computational theory.

I am really speaking here about operators, of course. Operators provide a mathematical model for describing dynamical systems. “Ensemble” (this usage of the word, by the way, is due to Albert Einstein) refers to a collection of such operators that share some common statistical properties.

To be more precise, Montgomery’s area of interest was the so-called “class number problem,” of which there is a very accessible account in Keith Devlin’s book, Mathematics: The New Golden Age (Columbia University Press, 1999).

Harold Diamond is a number theorist. He is currently Professor of Mathematics in the University of Illinois at Urbana-Champaign.

Sarvadaman Chowla, 1907–1995. A fine number theorist, mainly at the University of Colorado.

The standard introductory text on random matrix theory is Madan Lal Mehta’s Random Matrices and the Statistical Theory of Energy Levels (1991. New York: Academic Press).

Dyson was in fact another Trinity man, having attended that college in the early 1940s. He recalls that Hardy, at that time slipping into his terminal depression, was “not encouraging.”

This raises the interesting question of the degree to which these are really theorems. A result that assumes the truth of the RH is technically, it seems to me, a hypothesis itself—or perhaps a sub-hypothesis, but at any rate not a proper theorem. Considering, in fact, that mathematics is supposed to be the most precise of disciplines, mathematicians are not very consistent about the use of terms like “conjecture,” “hypothesis,” and “theorem.” Why, for example, is the RH a “hypothesis,” not a “conjecture”? I don’t know, and I haven’t found anyone who can tell me. These remarks seem, on a cursory examination, to apply in languages other than English, too. The German for “the Riemann Hypothesis,” by the way, is Die Riemannsche Vermutung, from the verb vermuten—“to surmise.”

Professor of Physics at Bristol University in England. Berry was elevated to the knightage in the Queen’s Birthday Honors of June, 1996, becom-

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.

The Cray-1 was supplemented by a Cray X-MP at some point in the late 1980s.

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.

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.

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).

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.

Though the word “chaos” was not applied to these theories until 1976, when physicist James Yorke first coined it. James Gleick’s 1987 bestseller Chaos: Making a New Science remains the best guide to chaos

theory for the layman … unless you count Tom Stoppard’s 1993 play Arcadia.

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).

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.

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.

“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.

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).

A good article on this topic is “Is π Normal?” by Stan Wagon, in Mathematical Intelligencer, Vol. 7, No. 3.

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.”

Mathematics and Plausible Reasoning (1954).

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.

I should perhaps say, for the benefit of any reader so fired up by my exposition as to be on the point of running out and buying a math software package, that very strong opinions are held about the relative merits of the different packages, along the lines of the evergreen PC/ Macintosh debate, with Stephen Wolfram, who created Mathematica, playing the part of Bill Gates. As a mere journalist, I consider myself hors de combat in this war. I am certainly not propagandizing on behalf of Mathematica. It was the first math software package that came to my attention, and it is the only one I have ever used. It has always done what I asked it to do. Sometimes, to be sure, I had to tweak it a little (see Note 128), but I never knew a software package that didn’t need tweaking now and again.

It has no direct bearing on the argument here, but I can’t resist adding, as a matter of interest, that one of the most famous theorems in complex function theory concerns entire functions. The theorem was stated and proved by Émile Picard (1856–1941). Picard’s Theorem says that if an entire function takes more than one value—if, that is, it is not merely

a flat constant function—then it takes every value, with at most one exception. For e^z, the exception is 0.

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 .

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.

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 .

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….

Ulrike’s pages on the University of Ulm website have a photograph of her standing next to Bernhard Riemann’s memorial stone in Selasca, Italy.

Professor of Applied Mathematics at the University of Bristol, England. Keating has worked closely with Sir Michael Berry on the physical aspects of the RH.

“The zeros of Mellin transforms of Hermite functions have real part one-half,” (1986). Bump’s collaborator in the proof was one E. K.-S. Ng, otherwise unknown to me.

So it seems to me. One of the professional mathematicians who looked over my manuscript expressed frank disbelief at this, though. The idea that one might be able to make money by doing mathematics is extremely difficult for mathematicians to take seriously.

Professor of Pure Mathematics, University of Wales, Cardiff.

Here is the chain of events in barest outline. The method adopted for Principia Mathematica offered no guarantee against flaws, like the flaw Russell had spotted in Frege’s work. Hilbert’s “metamathematics” program tried to encompass both logic and mathematics in a more water-proof symbolism. This inspired the work of Kurt Gödel and Alan Turing. Gödel proved important theorems by attaching numbers to Hilbert-type symbols; Turing coded both instructions and data as arbitrary numbers in his “Turing machine” concept. Picking up on this idea, John von Neumann developed the stored-program concept on which all modern software is based, that code and data can be represented in the same way in a computer’s memory….

In a letter to his brother dated June 26, 1854, he mentioned a recurrence of mein altes Übel—“my old malady”—brought on by a spell of bad weather.

In the modern municipality of Verbania.

Weender Chaussee has since been renamed Bertheaustrasse.