john derbyshire - prime obsession bernhard riemann and the greatest unsolved problem in mathemati

For the complete solution, however, of the problems set us by Riemann’s paper “On the Number of Prime Numbers Less Than a Given Quantity,” it still remains to prove the correctness of an

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P RIME O BSESSION

The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public Joseph Henry was one

of the early founders of the National Academy of Sciences and a leader in early American science.

Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions.

Library of Congress Cataloging-in-Publication Data

Derbyshire, John.

Prime obsession : Bernhard Riemann and the greatest unsolved problem

in mathematics / John Derbyshire.

Copyright 2003 by John Derbyshire All rights reserved.

Printed in the United States of America.

For Rosie

Prologue ix

Part I The Prime Number Theorem 1 Card Trick 3

2 The Soil, the Crop 19

3 The Prime Number Theorem 32

4 On the Shoulders of Giants 48

5 Riemann’s Zeta Function 63

6 The Great Fusion 82

7 The Golden Key, and an Improved Prime Number Theorem 99

8 Not Altogether Unworthy 118

9 Domain Stretching 137

10 A Proof and a Turning Point 151

Part II The Riemann Hypothesis

11 Nine Zulu Queens Ruled China 169

12 Hilbert’s Eighth Problem 184

13 The Argument Ant and the Value Ant 201

14 In the Grip of an Obsession 223

15 Big Oh and Möbius Mu 238

16 Climbing the Critical Line 252

17 A Little Algebra 265

18 Number Theory Meets Quantum Mechanics 280

19 Turning the Golden Key 296

20 The Riemann Operator and Other Approaches 312

21 The Error Term 327

22 Either It’s True, or Else It Isn’t 350

Epilogue 362

Notes 365

Appendix: The Riemann Hypothesis in Song 393

Picture Credits 405

Index 407

In August 1859, Bernhard Riemannwas made a corresponding member of the Berlin Academy, a greathonor for a young mathematician (he was 32) As was customary onsuch occasions, Riemann presented a paper to the Academy giving anaccount of some research he was engaged in The title of the paperwas: “On the Number of Prime Numbers Less Than a Given Quan-tity.” In it, Riemann investigated a straightforward issue in ordinaryarithmetic To understand the issue, ask: How many prime numbersare there less than 20? The answer is eight: 2, 3, 5, 7, 11, 13, 17, and 19.How many are there less than one thousand? Less than one million?

Less than one billion? Is there a general rule or formula for how many

that will spare us the trouble of counting them?

Riemann tackled the problem with the most sophisticated ematics of his time, using tools that even today are taught only inadvanced college courses, and inventing for his purposes a math-ematical object of great power and subtlety One-third of the wayinto the paper, he made a guess about that object, and then remarked:

math-One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of

my investigation.

That casual, incidental guess lay almost unnoticed for decades.Then, for reasons I have set out to explain in this book, it graduallyseized the imaginations of mathematicians, until it attained the sta-tus of an overwhelming obsession

The Riemann Hypothesis, as that guess came to be called, mained an obsession all through the twentieth century and remainsone today, having resisted every attempt at proof or disproof Indeed,the obsession is now stronger than ever since other great old openproblems have been resolved in recent years: the Four-Color Theo-rem (originated 1852, proved in 1976), Fermat’s Last Theorem (origi-nated probably in 1637, proved in 1994), and many others less wellknown outside the world of professional mathematics The RiemannHypothesis is now the great white whale of mathematical research.The entire twentieth century was bracketed by mathematicians’preoccupation with the Riemann Hypothesis Here is David Hilbert,one of the foremost mathematical intellects of his time, addressingthe Second International Congress of Mathematicians at Paris in Au-gust 1900:

re-Essential progress in the theory of the distribution of prime bers has lately been made by Hadamard, de la Vallée Poussin, von Mangoldt and others For the complete solution, however, of the problems set us by Riemann’s paper “On the Number of Prime Numbers Less Than a Given Quantity,” it still remains to prove the correctness of an exceedingly important statement of Riemann, viz .

num-There follows a statement of the Riemann Hypothesis A dred years later, here is Phillip A Griffiths, Director of the Institutefor Advanced Study in Princeton, and formerly Professor of Math-

hun-P ROLOGUE xi

ematics at Harvard University He is writing in the January 2000 issue

of American Mathematical Monthly, under the heading: “Research

Challenges for the 21st Century”:

Despite the tremendous achievements of the 20th century, dozens

of outstanding problems still await solution Most of us would ably agree that the following three problems are among the most challenging and interesting.

prob-The Riemann Hypothesis prob-The first is the Riemann Hypothesis,

which has tantalized mathematicians for 150 years .

An interesting development in the United States during the lastyears of the twentieth century was the rise of private institutes formathematical research, funded by wealthy math enthusiasts Both theClay Mathematics Institute (founded by Boston financier Landon T.Clay in 1998) and the American Institute of Mathematics (established

in 1994 by California entrepreneur John Fry) have targeted the mann Hypothesis The Clay Institute has offered a prize of one mil-lion dollars for a proof or a disproof; the American Institute of Math-ematics has addressed the Hypothesis with three full-scaleconferences (1996, 1998, and 2002), attended by researchers from allover the world Whether these new approaches and incentives willcrack the Riemann Hypothesis at last remains to be seen

Rie-Unlike the Four-Color Theorem, or Fermat’s Last Theorem, theRiemann Hypothesis is not easy to state in terms a nonmathematiciancan easily grasp It lies deep in the heart of some quite abstruse math-ematical theory Here it is:

The Riemann Hypothesis

All non-trivial zeros of the zeta function

have real part one-half

To an ordinary reader, even a well-educated one, who has had noadvanced mathematical training, this is probably quite incomprehen-

sible It might as well be written in Old Church Slavonic In this book,

as well as describing the history of the Hypothesis, and some of thepersonalities who have been involved with it, I have attempted tobring this deep and mysterious result within the understanding of ageneral readership, giving just as much mathematics as is needed tounderstand it

* * * * *

The plan of the book is very simple The odd-numbered chapters

(I was going to make it the prime-numbered, but there is such a thing

as being too cute) contain mathematical exposition, leading the

reader, gently I hope, to an understanding of the Riemann esis and its importance The even-numbered chapters offer historicaland biographical background matter

Hypoth-I originally intended these two threads to be independent, so thatreaders who don’t like equations and formulae could read only theeven-numbered chapters while readers who did not care for history

or anecdote could just read the odd-numbered ones I did not quitemanage to hold to this plan all the way through, and I now doubt that

it can be done with a subject so intricate Still, the basic pattern wasnot altogether lost There is much more math in the odd-numberedchapters, and much less in the even-numbered ones, and you are, ofcourse, free to try reading just the one group or the other I hope,though, that you will read the whole book

I have aimed this book at the intelligent and curious butnonmathematical reader That statement, of course, raises a number

of questions What do I mean by “nonmathematical?” How muchmath knowledge have I assumed my readers possess? Well, everybody

knows some math Probably most educated people have at least an inkling of what calculus is all about I think I have pitched my book to

the level of a person who finished high school math satisfactorily andperhaps went on to a couple of college courses My original goal was,

in fact, to explain the Riemann Hypothesis without using any calculus

P ROLOGUE xiii

at all This proved to be a tad over-optimistic, and there is a very

small quantity of very elementary calculus in just three chapters, plained as it goes along

ex-Pretty much everything else is just arithmetic and basic algebra:

multiplying out parentheses like (a + b) × (c + d), or rearranging equations so that S = 1 + xS becomes S = 1 ⁄ (1 – x) You will also need

a willingness to take in the odd shorthand symbols mathematiciansuse to spare the muscles of their writing hands I claim at least thismuch: I don’t believe the Riemann Hypothesis can be explained us-ing math more elementary than I have used here, so if you don’t un-derstand the Hypothesis after finishing my book, you can be prettysure you will never understand it

* * * * *

Various professional mathematicians and historians of ematics were generous with their help when I approached them I amprofoundly grateful to the following for their time, freely given, fortheir advice, sometimes not taken, for their patience in dealing with

math-my repetitive dumb questions, and in one case for the hospitality ofhis home: Jerry Alexanderson, Tom Apostol, Matt Brin, Brian Conrey,Harold Edwards, Dennis Hejhal, Arthur Jaffe, Patricio Lebeuf,Stephen Miller, Hugh Montgomery, Erwin Neuenschwander, AndrewOdlyzko, Samuel Patterson, Peter Sarnak, Manfred Schröder, UlrikeVorhauer, Matti Vuorinen, and Mike Westmoreland Any gross errors

in this book’s math are mine, not theirs Brigitte Brüggemann andHerbert Eiteneier helped plug the gaps in my German Commissions

from my friends at National Review, The New Criterion, and The

Washington Times allowed me to feed my children while working on

this book Numerous readers of my online opinion columns helped

me understand what mathematical ideas give the most difficulty tononmathematicians

Along with these acknowledgments goes an approximately equalnumber of apologies The topic this book deals with has been under

intensive investigation by some of the best minds on our planet for ahundred years In the space available to me, and by the methods ofexposition I have decided on, it has proved necessary to omit entirelarge regions of inquiry relevant to the Riemann Hypothesis You willfind not one word here about the Density Hypothesis, the approxi-mate functional equation, or the whole fascinating issue—just re-cently come to life after long dormancy—of the moments of the zetafunction Nor is there any mention of the Generalized Riemann Hy-pothesis, the Modified Generalized Riemann Hypothesis, the Ex-tended Riemann Hypothesis, the Grand Riemann Hypothesis, theModified Grand Riemann Hypothesis, or the Quasi-Riemann Hy-pothesis.

Even more distressing, there are many workers who have toiledaway valiantly in these vineyards for decades, but whose names areabsent from my text: Enrico Bombieri, Amit Ghosh, Steve Gonek,Henryk Iwaniec (half of whose mail comes to him addressed as

“Henry K Iwaniec”), Nina Snaith, and many others My sincereapologies I did not realize, when starting out, what a vast subject Iwas taking on This book could easily have been three times, or thirtytimes, longer, but my editor was already reaching for his chainsaw.And one more acknowledgment I hold the superstitious beliefthat any book above the level of hired drudge work—any book writ-ten with care and affection—has a presiding spirit By that, I only

mean to say that a book is about some one particular human being,

who is in the author’s mind while he works, and whose personalitycolors the book (In the case of fiction, I am afraid that all too oftenthat human being is the author himself.)

The presiding spirit of this book, who seemed often to be ing over my shoulder as I wrote, whom I sometimes imagined I heardclearing his throat shyly in an adjoining room, or moving arounddiscreetly behind the scenes in both my mathematical and historicalchapters, has been Bernhard Riemann Reading him, and readingabout him, I developed an odd mixture of feelings for the man: greatsympathy for his social awkwardness, wretched health, repeated be-

dedi-John DerbyshireHuntington, New YorkJune 2002

THE PRIME

NUMBER THEOREM

I

C ARD T RICK

Like many other performances, thisone begins with a deck of cards

Take an ordinary deck of 52 cards, lying on a table, all four sides

of the deck squared away Now, with a finger slide the topmost cardforward without moving any of the others How far can you slide itbefore it tips and falls? Or, to put it another way, how far can youmake it overhang the rest of the deck?

FIGURE 1-1

I.

The answer, of course, is half a card length, as you can see inFigure 1-1 If you push it so that more than half the card overhangs, itfalls The tipping point is at the center of gravity of the card, which ishalfway along it.

Now let’s go a little further With that top card pushed out half itslength—that is, to maximum overhang—over the second one, pushthat second card with your finger How much combined overhangcan you get from these top two cards?

The trick is to think of these top two cards as a single unit Where

is the center of gravity of this unit? Well, it’s halfway along the unit,which is altogether one and a half cards long; so it’s three-quarters of

a card length from the leading edge of the top card (see Figure 1-2).The combined overhang is, therefore, three-quarters of a card length.Notice that the top card still overhangs the second one by half a cardlength You moved the top two cards as a unit

FIGURE 1-2

If you now start pushing the third card to see how much you canincrease the overhang, you find you can push it just one-sixth of acard length Again, the trick is to see the top three cards as a singleunit The center of gravity is one-sixth of a card length back from theleading edge of the third card (see Figure 1-3)

C ARD T RICK 5

FIGURE 1-3

In front of this point is one-sixth of the third card, a sixth plus aquarter of the second card, and a sixth plus a quarter plus a half of thetop card, making a grand total of one and a half cards

16

16

14

16

14

1

12+ + + + +  =

FIGURE 1-4

That’s half of three cards—the other half being behind the tippingpoint Here’s what you have after pushing that third card as far as itwill go (see Figure 1-4)

The total overhang now is a half (from the top card) plus a ter (from the second) plus a sixth (from the third) This is a total ofeleven twelfths of a card Amazing!

quar-Can you get an overhang of more than one card? Yes you can.The very next card—the fourth from the top—if pushed forwardcarefully, gives another one-eighth of a card length overhang I’m notgoing to do the arithmetic; you can trust me, or work it out as I didfor the first three cards Total overhang with four cards: one-half plusone-quarter plus one-sixth plus one-eighth, altogether one and one-twenty-fourth card lengths (see Figure 1-5)

18

110

112

114

116

1102+ + + + + + + + +L

for the 51 cards you push (No point pushing the very bottom one.)This comes out to a shade less than 2.25940659073334 So you have atotal overhang of more than two and a quarter card lengths! (SeeFigure 1-6.)

FIGURE 1-6

I was a college student when I learned this It was summer tion and I was prepping for the next semester’s work, trying to getahead of the game To help pay my way through college I used tospend summer vacations as a laborer on construction sites, work thatwas not heavily unionized at the time in England The day after Ifound out about this thing with the cards I was left on my own to dosome clean-up work in an indoor area where hundreds of large,square, fibrous ceiling tiles were stacked I spent a happy couple ofhours with those tiles, trying to get a two and a quarter tile overhangfrom 52 of them When the foreman came round and found me deep

vaca-in contemplation of a great wobblvaca-ing tower of ceilvaca-ing tiles, I suppose

C ARD T RICK 7

his worst fears about the wisdom of hiring college students must havebeen confirmed

II. One thing mathematicians like to do, and find very fruitful, is

extrapolation—taking the assumptions of a problem and stretching

them to cover more ground

I assumed in the above problem that we had 52 cards to workwith We found that we could get a total overhang of better than twoand a quarter cards

Why restrict ourselves to 52 cards? Suppose we had more? A

hun-dred cards? A million? A trillion? Suppose we had an unlimited

sup-ply of cards? What’s the biggest possible overhang we could get?First, look at the formula we started to develop With 52 cards thetotal overhang was

1

2

14

16

18

110

112

114

116

1102

13

14

15

16

17

18

151+ + + + + + + + +

13

14

15

16

17

18

199+ + + + + + + + +

14

15

16

17

18

1999999999999+ + + + + + + + +



That’s a lot of arithmetic; but mathematicians have shortcuts for thiskind of thing, and I can tell you with confidence that the total over-hang with a hundred cards is a tad less than 2.58868875882, while for

a trillion cards it is a wee bit more than 14.10411839041479

These numbers are doubly surprising The first surprise is thatyou can get a total overhang of more than 14 full card lengths, eventhough you need a trillion cards to get it Fourteen card lengths ismore than four feet, with standard playing cards The second sur-prise, when you start thinking about it, is that the numbers aren’tbigger Going from 52 cards to 100 got us only an extra one-third of acard overhang (a bit less than one-third, in fact) Then going all theway to a trillion—a stack of a trillion standard playing cards would

go most of the way from the Earth to the Moon—gained us onlyanother 111

as big as you want, if you’re willing to use unimaginably large bers of cards A million-card overhang? Sure, but the number of cardsyou need now is so huge it would need a fair-sized book just to print

num-it in—num-it has 868,589 dignum-its

III. The thing to concentrate on here is that expression inside theparentheses

1 12

13

14

15

16

17+ + + + + + +L

This is what mathematicians call a series, addition of terms

continu-ing indefinitely, where the terms follow some logical progression.Here the terms 1, 1

7+… is sufficiently important that

mathematicians have a name for it It is called the harmonic series.

What I have stated above amounts to this: by adding enoughterms of the harmonic series, you can get a total as big as you please.The total has no limit

A crude, but popular and expressive, way to say this is: the monic series adds up to infinity

har-1 12

13

14

15

16

17+ + + + + + + = ∞L

Well-brought-up mathematicians are taught to sniff at expressionslike that; but so long as you know the pitfalls of using them I thinkthey are perfectly all right Leonhard Euler, one of the half-dozengreatest mathematicians who ever lived, used them all the time withvery fruitful results However, the proper mathematical term of art is:

The harmonic series is divergent.

Well, I have said this, but can I prove it? Everybody knows that inmathematics you must prove every result by strict logic Here we have

a result: the harmonic series is divergent How do you prove it?The proof is, in fact, rather easy and depends on nothing morethan ordinary arithmetic It was produced in the late Middle Ages by

a French scholar, Nicole d’Oresme (ca 1323-1382) D’Oresme

pointed out that 1

D’Oresme’s proof of the divergence of the harmonic series seems

to have been mislaid for several centuries Pietro Mengoli proved theresult all over again in 1647, using a different method; then, fortyyears later, Johann Bernoulli proved it using yet another method; andshortly after that, Johann’s elder brother Jakob produced a proof by afourth method Neither Mengoli nor the Bernoullis seem to have beenaware of d’Oresme’s fourteenth-century proof, one of the barelyknown masterpieces of medieval mathematics D’Oresme’s proof re-mains the most straightforward and elegant of all the proofs, though,and is the one usually given in textbooks today

IV. The amazing thing about series is not that some of them aredivergent, but that any of them are not If you add together an infin-ity of numbers, you expect to get an infinite result, don’t you? Thefact that you sometimes don’t can be easily illustrated

Take an ordinary ruler marked in quarters, eighths, sixteenths,and so on (the more “so on” the better—I’ve shown a ruler marked insixty-fourths) Hold a sharp pencil point at the very first mark on theruler, the zero Move the pencil one inch to the right The pencil point

is now on the one-inch mark and you have moved it a total of oneinch (see Figure 1-7)

64ths

FIGURE 1-7

18

116

132

164

which is, as you can see, 163

64 Clearly, if you could go on like this,halving the distance each time, you would get closer and closer to thetwo-inch mark You would never quite reach it; but there is no limit

to how close you could get You could get to within a millionth of aninch of it; or a trillionth; or a trillion trillion trillion trillion trilliontrillion trillion trillion trillionth We can express this fact as

1 1

2

14

18

116

132

164

har-an infinite number of terms har-and got infinity Here I am adding up har-an

infinite number of terms and getting 2 The harmonic series is

diver-gent This one is converdiver-gent.

The harmonic series has its charms, and it stands at the center ofthe topic this book addresses—the Riemann Hypothesis Generallyspeaking, however, mathematicians are more interested in conver-gent series than divergent ones

V. Suppose that instead of moving one inch to the right, then ahalf-inch to the right, then a quarter-inch to the right, and so on, Idecided to alternate directions: an inch to the right, a half-inch to theleft, a quarter-inch to the right, an eighth-inch to the left.… Afterseven moves I’d be at the point shown in Figure 1-10

18

116

132

164

which is 43

64 In fact, it’s rather easy to show—I’ll prove it in a laterchapter—that if you keep on adding and subtracting to infinity youget

1 1

2

14

18

116

132

164

1128

23

Expression 1-2

VI. Now, suppose that instead of starting out with a ruler marked

in halves, quarters, eighths, sixteenths, and so on, I have a rulermarked in thirds, ninths, twenty-sevenths, eighty-firsts, and so on Inother words, instead of halves, halves of halves, halves of halves ofhalves … I have thirds, thirds of thirds, thirds of thirds of thirds, and

so on And suppose I do an exercise similar to the first one, move thepencil along one inch, then a third of an inch, then a ninth, then atwenty-seventh (see Figure 1-11)

FIGURE 1-11

I don’t think it’s too hard to see that if you continue forever, you end

up moving right a total 11

2 inches as shown in Expression 1-3 That is,

1 1

3

19

127

181

1243

1729

1

12

Expression 1-3And of course, I can do the alternating movement with this new ruler,too: right one inch, left a third, right a ninth, left a twenty-seventh,and so on (see Figure 1-12)

81sts

FIGURE 1-12The math of Expression 1-4 is not so visually obvious, but it’s a factthat

C ARD T RICK 15

1 1

3

19

127

181

1243

1729

12187

34

Expression 1-4

So here we have four convergent series, the first (Expression 1-1)

creeps closer and closer to 2 from the left, the second (Expression 2) closes in on 2

1-3 from left and right alternately, the third (Expression1-3) creeps closer and closer to 11

2 from the left, the fourth sion 1-4) closes in on 3

(Expres-4 from left and right alternately Before that, I

showed one divergent series, the harmonic series.

VII. When reading math, it is important to know where in mathyou are—what region of this vast subject you are exploring The par-ticular zone these infinite series dwell in is what mathematicians call

analysis Analysis used, in fact, to be thought of as the study of the

infinite, that is, the infinitely large, and of the infinitesimal, the nitely small When Leonhard Euler—of whom I shall write muchmore later—published the first great textbook of analysis in 1748, he

infi-called it Introductio in analysin infinitorum: “Introduction to the

Analysis of the Infinite.”

The notions of the infinite and the infinitesimal created seriousproblems in math during the early nineteenth century, though, andeventually they were swept away altogether in a great reform Modernanalysis does not admit these concepts They linger on in the vocabu-lary of mathematics, and I shall make free use of the word “infinity”

in this book This usage, however, is only a convenient and tive shorthand for more rigorous concepts Every mathematical state-ment that contains the word “infinity” can be reformulated withoutthat word

imagina-When I say that the harmonic series adds up to infinity, what I

really mean is that given any number S, no matter how large, the sum

of the harmonic series eventually exceeds S See?—No “infinity.” The

whole of analysis was rewritten in this kind of language in the middlethird of the nineteenth century Any statement that can’t be so rewrit-ten is not allowed in modern mathematics Nonmathematical peoplesometimes ask me, “You know math, huh? Tell me something I’vealways wondered, What is infinity divided by infinity?” I can onlyreply, “The words you just uttered do not make sense That was not amathematical sentence You spoke of ‘infinity’ as if it were a number.It’s not You may as well ask, ‘What is truth divided by beauty?’ I have

no clue I only know how to divide numbers ‘Infinity,’ ‘truth,’

‘beauty’—those are not numbers.”

What is a modern definition of analysis, then? I think the study of

limits will do for my purposes here The concept of a limit is at the

heart of analysis All of calculus, for example, which forms the largestpart of analysis, rests on the idea of a limit

Consider the following sequence of numbers: 1

2378, for example, you get 11309769

5654884, which is2.000000176838287… We say that the limit of the sequence is 2.Here is another case: 4

1 8 3 32 9 128 45 768 225 4608 1575 36864 11025 294912 99225

2 2

( ) , 11

3 3

Notice that all of these are sequences, just strings of numbers rated by commas They are not series, where the numbers are actually

sepa-added up From the point of view of analysis, however, a series is just

a sequence in thin disguise The statement “The series 1 + 1 + 1 + 1

, , , , , ,…converges to 2.” The fourth term ofthe sequence is the sum of the first four terms of the series, and so on.

(The term of art for this kind of sequence is the sequence of partial

sums.) Similarly, of course, the statement, “The harmonic series

di-verges” is equivalent to: “The sequence 1 1 11 2 2 2

2 5 6 1 12 17 60 27 60

num-say it has the limit a, I mean that no matter how tiny a number x you

pick, from some point on, every number in the sequence differs from

a by less than x If you choose to say: “The sequence is infinite,” or:

“The limit of the N-th term, when N goes to infinity, is a,” you are free

to do so, as long as you understand that these are just loose and venient ways of speaking

con-VIII. The traditional division of mathematics into subdisciplines is

as follows

Sample theorem: If you subtract an odd number from an evennumber you get an odd number

curves, and three-dimensional objects Sample theorem: Theangles of a triangle on a flat surface add up to 180 degrees

■ Algebra—The use of abstract symbols to represent

mathemati-cal objects (numbers, lines, matrices, transformations), andthe study of the rules for combining those symbols Sample

theorem: For any two numbers x and y, (x + y) × (x − y) = x2−

y2

■ Analysis—The study of limits Sample theorem: The harmonic

series is divergent (that is, it increases without limit)

Modern mathematics contains much more than that, of course

It includes set theory, for example, created by Georg Cantor in 1874,and “foundations,” which another George, the Englishman GeorgeBoole, split off from classical logic in 1854, and in which the logicalunderpinnings of all mathematical ideas are studied The traditionalcategories have also been enlarged to include big new topics—geom-etry to include topology, algebra to take in game theory, and so on.Even before the early nineteenth century there was considerable seep-age from one area into another Trigonometry, for example, (the wordwas first used in 1595) contains elements of both geometry and alge-bra Descartes had in fact arithmetized and algebraized a large part ofgeometry in the seventeenth century, though pure-geometric dem-onstrations in the style of Euclid were still popular—and still are—for their clarity, elegance, and ingenuity

The fourfold division is still a good rough guide to finding yourway around mathematics, though It is a good guide, too, for under-standing one of the greatest achievements of nineteenth-centurymath, what I shall later call “the great fusion”—the yoking of arith-metic to analysis to create an entirely new field of study, analytic num-ber theory Permit me to introduce the man who, with one singlepublished paper of eight and a half pages, got analytic number theoryoff the ground and flying

T HE S OIL , THE C ROP

Bernhard Riemann He left no record of his inner life, other thanwhat can be deduced from his letters His friend and contemporary,Richard Dedekind, was the only person close to him who wrote adetailed memoir; but that was a mere 17 pages and revealed little.What follows, therefore, cannot hope to capture Riemann, but I hope

it will at least leave him more than a mere name in the reader’s mind

I have reduced his academic career to a brief sketch in this chapter Ishall describe it in much more detail in Chapter 8

First, let me set the man in his time and place

II. Supposing that their Revolution had left the French disorganizedand ineffective, and disturbed by its republican and antimonarchicalideals, France’s enemies moved to take advantage of the situation In

1792 a huge force of mainly Austrian and Prussian troops, but whichincluded 15,000 emigré French, advanced on Paris To their surprise,

I.

the army of revolutionary France took a stand at the village of Valmy,engaging the invaders in an artillery duel fought in thick fog on Sep-

tember 20 of that year Edward Creasy, in his classic Fifteen Decisive

Battles of the World, calls this the Battle of Valmy Germans call it the

Cannonade of Valmy By either name it is a convenient marker for thebeginning of the succession of wars that occupied Europe for the next

23 years The Napoleonic Wars is the usual name given to these events;though it would be logical, if the expression were not already spokenfor, to put them all under the heading First World War, since theyincluded engagements in both the Americas and the Far East When

it all ended at last, with a peace treaty worked out at the Congress ofVienna (June 8, 1815), Europe settled into a long period, almost acentury, of relative peace

Hanover Berlin

Leipzig Anhalt

Dresden

Breselenz Bremen

l b e

Northwest Germany after 1815 Note that Hanover (the state) is in two pieces; both Hanover (the city) and Göttingen belong to it Prussia is in two large pieces and some smaller ones; both Berlin and Cologne are Prussian

cities Brunswick is in three pieces.

T HE S OIL , THE C ROP 21

One consequence of the treaty was a modest tidying up of theGerman peoples in Europe Before the French Revolution a German-speaking European might have been a citizen of Hapsburg Austria(in which case he was probably a Catholic) or of the Kingdom ofPrussia (making him more likely a Protestant) or of any one of threehundred-odd petty principalities scattered across the map of what wenow call Germany He might also have been a subject of the king ofFrance, or of the king of Denmark, or a citizen of the Swiss Confed-eration “Tidying up” is a relative term—there was enough untidinessleft over to occasion several minor wars, and to contribute to the twogreat conflicts of the twentieth century Austria still had her empire(which included great numbers of non-Germans: Hungarians, Slavs,Romanians, Czechs, and so on); Switzerland, Denmark, and Francestill included German speakers It was a good start, though The threehundred-odd entities that comprised eighteenth-century Germanywere consolidated into 34 sovereign states and 4 free cities, and theircultural unity was recognized by the creation of a German Confed-eration

The largest German states were still Austria and Prussia Austria’spopulation was about 30 million, only 4 million of them Germanspeakers Prussia had about 15 million citizens, most of them Ger-man speakers Bavaria was the only other German state with a popu-lation over 2 million Only four others had more than a million: thekingdoms of Hanover, Saxony, and Württemberg, and the GrandDuchy of Baden

Hanover was something of an oddity in that, although a dom, its king was hardly ever present The reason for this was that, forcomplicated dynastic reasons, he was also king of England The firstfour of what English people call the “Hanoverian kings” were allnamed George,1 and the fourth was on the throne in 1826, when thecentral character in the story of the Riemann Hypothesis firstappeared

king-III. Georg Friedrich Bernhard Riemann was born on September

17, 1826, in the village of Breselenz in the eastern salient of the dom of Hanover This part of the kingdom is known as Wendland,

King-“Wend” being an old German word for the Slavic-speaking peoplesthey encountered Wendland was the furthest west reached by thegreat Slavic advance of the sixth century The name “Breselenz” itselfderives from the Slavic word for “birch-tree.” Slavic dialects and folk-lore survived into modern times—the philosopher Leibnitz (1646−1716) promoted research into them—but from the late Middle Agesonward German immigrants moved into Wendland and byRiemann’s time the population was pretty solidly German

Wendland was, and still is, something of a backwater With only

110 inhabitants per square mile, it is the most thinly populated trict in its modern region, Lower Saxony There is little industry andfew large towns The mighty Elbe—it is about 250 yards wide here—flows just 7 miles from Breselenz and was the principal connectionwith the world beyond until modern times In the nineteenth centurysailing ships and barges carried timber and agricultural producedown to Hamburg from Central Europe, returning with coal and in-dustrial goods During the recent decades of division, the Wendlandstretch of the Elbe was part of the border between East and WestGermany, a fact that did nothing to help local development It is aflat, dull countryside of farm, heath, marsh, and thin woodland, prone

dis-to flooding There was a serious flood in 1830 that must have beenthe first great external event of Bernhard Riemann’s childhood.2Riemann’s father, Friedrich Bernhard Riemann, was a Lutheranminister and a veteran of the wars against Napoleon He was alreadymiddle-aged when he married Charlotte Ebell Bernhard was theirsecond child and seems to have been especially close to his older sis-ter, Ida—he named his own daughter after her Four more childrenfollowed, a boy and three girls With today’s standard of living, which

of course we take for granted, it is difficult to imagine the hardshipsthat faced a country parson, well into his middle years, with a wife

T HE S OIL , THE C ROP 23

and six children to support, in a poor and undeveloped region of amiddling country in the early nineteenth century Of the six Riemannchildren, only Ida lived a normal life span The others all died young,probably in part from poor nutrition Riemann’s mother, too, diedyoung, before her children were grown

Poverty aside, it needs an effort of imagination for us, living andworking in a modern economy, to grasp the sheer difficulty of find-ing a job in those times and circumstances Outside large cities themiddle class barely existed There was a scattering of merchants, par-sons, schoolteachers, physicians, and government officials Everyoneelse who did not own land was a craftsman, a domestic servant, or apeasant The only respectable employment for women was as gov-ernesses; otherwise they relied on their husbands or male familymembers for support

When Bernhard was still an infant, his father took up a new tion as minister in Quickborn, a few miles from Breselenz, and closer

posi-to the great river Quickborn is still, posi-today, a sleepy village of framed houses and mostly unpaved streets bordered by massive, an-cient oak trees This place, even smaller than Breselenz, remained thefamily home until the elder Riemann died in 1855 It was the center

timber-of Bernhard’s emotional world until he was almost 30 years old Heseems to have returned there at every opportunity to be amongst hisfamily, the only surroundings in which he ever felt at ease

In reading of Riemann’s life, therefore, one must set it all against

a backdrop of this environment, the environment of his home andupbringing, which he cherished, and for which, when away from it,

he yearned The flat, damp countryside; the draughty house lit only

by oil lamps and candles, ill-heated in winter and ill-ventilated insummer; long spells of sickness among siblings who themselves werenever quite well (they seem all to have suffered from tuberculosis);the tiny and monotonous social round of a parson’s family in a re-mote village; the inadequate and unbalanced diet on the stodgy side

of a stodgy national cuisine (“For a long time he suffered from