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

I. The Chinese word Taiye (pronounced “tie-yeah”) translates literally as “ultimate grandpa.” It is the title given in my wife’s family to her paternal grandfather. Visiting China in the summer of 2001, our first duty was to call on Taiye. The family is immensely proud of him, for he has lived to age 97 in good health and with a clear head. “Ninety-seven years old now!” they all told me. “You should see him!” Well, I did see him—a fine cheerful Buddha of a man, his face glowing ruddy and his mind still sharp. Whether he was actually 97 at the time is, however, an interesting point.

Taiye was born on the third day of the twelfth lunar month of the lunar year named yi si in the traditional “Heaven-Earth” year-numbering system. This day was December 28, 1905, on the Western calendar. Since my visit occurred early in July 2001, Taiye’s age at the time was, in the modern western reckoning, 95½ years and a few days. So why was everyone telling me that he was 97? Because in the old Chinese style, which Taiye clove to, he was one year old at birth, and another one year old when each Lunar New Year rolled round— which one did, on January 24, 1906, by our calendar, 27 days after his

birth. Not yet one month in the world, and he was already two years old! Thus, when the Lunar New Year arrived in 2001 (also on January 24, as it happened, though Lunar New Year can fall on any date between January 21 and February 20), Ultimate Grandpa hailed it as his 97th year.

There is nothing wrong with the logic behind this traditional Chinese system of reckoning age. You come into the world on a certain day. That day belongs to a certain year. Obviously, that is your first year. If, 28 days later, a new year dawns—well, that will be your second year. It all makes perfect sense. The only reason it seems odd is that in the matter of computing our ages, modern people (in China as well as in the West) have got accustomed to dealing with time as something to be measured. In Taiye’s young days, the Chinese thought of a person’s age as something to be counted.

II. This distinction between numbers for counting and numbers for measuring reaches deep into human habits of thought and speech. It is as if with one part of our minds we perceive the world as made up of distinct, solid objects that can be tallied; while with another, we see it as a collection of fabrics, grains, or fluids, to be divided up and measured. Keeping the two notions straight does not come easily. My son, six years old, still confuses “many” with “much.” To a friend, after the Christmas festivities, “How much presents did you get?”

Our perceptions of the world are mirrored in our languages. The English language takes the world to be mainly a countable place: one cow, two fishes, three mountains, four doors, five stars. Somewhat less frequently, our language takes the world to be measurable: one blade of grass, two sheets of paper, three head of cattle, four grains of rice, five gallons of gasoline. The words “blade,” “sheet,” “head,” “grain,” and “gallon,” though of course some of them have lives of their own, here are acting as units of measurement. The Chinese language, by contrast with English, takes very nearly the whole of cre-

ation to be measurable. One of the minor chores of learning Chinese is memorizing the right “measure word” (that is a precise translation of the Chinese grammatical term liang ci) for each noun: one head of cow, two sticks of fish, three plinths of mountain, four fans of door, five grains of star. In the entire Chinese language there are only two words that can always be let loose grammatically without a measure word: “day” and “year.” Everything else—cows, fishes, mountains, doors, stars—is a kind of stuff that must be divided up and measured out before we can talk about it.

The much/many confusion has occasioned much argument and many inconveniences. At the time of the millennium, for example, which most of us celebrated when the year 1999 turned into the year 2000, there was an irritating minority of dissidents who said we had it all wrong. The source of their complaint was the true fact that our common calendar was set up without a year zero. The first day of the year 1 C.E. was preceded by the last day of the year 1 B.C.E. This was because Dionysus Exiguus, the sixth-century monk who imposed a Christian year-numbering system on the months and days of Julius Caesar’s calendar, regarded years as countable things, just as our Taiye does. The first year of the Christian era was, therefore, to be the year 1, the second was to be year 2, and so on.

The error is easily understood. Look at a common desk ruler. (Not for the first time in this book. It is amazing how much math— even higher math—can be referred back to the marks on a $1.89 ruler.) Yes, there are 12 inches marked on it. Yes, you can count them: 1, 2, 3, 4, …, 12. Ah, but if you are an ant, and you begin walking from the left-hand end of the ruler to the right, and you have just covered the first half-inch, where are you? In the middle of the first inch? Yes. In the middle of inch 1, then? Sure, if you like. But what is the precise measure of the distance you have walked? Well, it is 0.5 inches. Since walking is a continuous process—since the ant will eventually traverse every point of the ruler—this is a much more interesting and important number for the mathematician. He therefore prefers to say that

you are halfway (that is, .5 of the way) through the zero-th inch, giving a position 0.5.

Modern people are sufficiently sophisticated about mathematics that they think like this quite naturally most of the time. That, in fact, was the source of confusion for those millennium complainers—or for the revelers late on the night of December 31, 1999, depending on which point of view you want to take. The complainers were saying: “If you measure the time from the starting instant of the common era to the very end of the year 1999, you only have 1,999 complete years. You should wait until 2,000 complete years have elapsed.” They were imposing measuring logic on a system created according to counting logic. The revelers, on the other hand, were saying: “Here comes year number 2,000! Whoopee!”—pure counting logic. Yet these same revelers might fall back on measuring logic if asked the age of their new baby: “Oh, he’s just half a year old.” Which is to say, his age is 0.5 years—measuring logic, at least by contrast with the traditional Chinese approach. (They might, of course, confuse the issue further by saying: “Six months.…”)

I once got into a mild controversy with the writer and word-lover William F. Buckley, Jr., about the word “data.” Is this a singular word or a plural word? The word originated with the Latin verb dare, “to give.” From this, by the ordinary processes of Latin grammar, a gerund (that is, a verbal noun) can be formed: datum, meaning “that which is given.” From this, in turn, you can make a plural: data—“those things that are given.” However, we are speaking English, not Latin. Plenty of Latin plurals are used as English singulars—agenda, for example. Nobody says “The agenda are prepared.” English is our language; if we borrow a word from another tongue, we can do with it as we please.

Having worked with data all my adult life, I know very well what it is. It is a stuff, made up of innumerable tiny particles, indistinguishable one from another—like rice, sand, or grass. This kind of stuff needs to be referred to, in English, with singular verb forms (“The rice is cooked”) or measure words. If you want to pluck out one par-

ticle and address it, you use a measure word: “A grain of rice,” “An item of data.” This is, in fact, how people who make a living handling data do speak, by instinct. Among people whose business is data, nobody ever says “One datum, two data.” If people did say this, nobody would understand them. The grammarians, however, still want us to say “The data are.…” I predict they will lose the battle eventually.

As a final example, one that used to puzzle me in my Church of England schooldays, consider the three days that Jesus Christ lay in his tomb before being resurrected, according to his own prophecy, “After three days I will rise again.” Three days? The Crucifixion occurred on a Friday—Good Friday. The Resurrection occurred on a Sunday. That’s 48 hours, measure-wise, but of course 3 days (Friday, Saturday, Sunday) counting-wise, which is how the Hellenized intellectuals who compiled the New Testament reckoned it.

III. The Riemann Hypothesis

All non-trivial zeros of the zeta function have real part one-half.

The Riemann Hypothesis was born out of an encounter, what my chapter heading calls a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.

To summarize the traditional categories of mathematics that I gave in a href="3.html#p200072a19970003001">Chapter 1.viii.

• Arithmetic—The study of whole numbers and fractions.

• Geometry—The study of figures in space.

• Algebra—The use of abstract symbols to represent mathemati

cal objects (numbers, lines, matrices, transformations), and the study of the rules for combining those symbols.

• Analysis—The study of limits.

This fourfold scheme was well established in people’s minds around 1800, and the great fusion I am going to describe in this chapter was a fusion of ideas which, until 1837, had lived separate lives under two of the above headings, arithmetic and analysis. This fusion created the discipline of analytic number theory.

We are quite blasé about these leaps of imagination nowadays, and perhaps a little better at them. Today, in fact, as well as analytic number theory, there is an algebraic number theory and a geometric number theory. (I shall introduce some algebraic number theory in Chapter 20.v.) In the 1830s, however, it was a very striking thing, to yoke together concepts from two areas previously thought to be unconnected. Before I can introduce you to the principal player in this phase of the story, though, I need to say a little more about those two disciplines he brought together.

IV. At the time I am speaking of—the early nineteenth century— analysis was still the newest and sexiest branch of math, where the greatest advances were being made and the keenest minds were working. We knew more about arithmetic, geometry, and algebra at the end of the nineteenth century than we did at the beginning, but we knew way more about analysis. At the opening of that century, in fact, the most fundamental concept of analysis, the concept of a limit, was not clearly understood even by the best minds. If you had asked Euler, or even the young Gauss,what analysis was all about, he would have said: “It is about the infinite and the infinitesimal.” If you had then asked Euler what, precisely, the infinite is, he would have had a coughing fit and left the room, or else opened a discussion about the meaning of “is.”

Analysis really dates from the invention of calculus by Newton and Leibnitz in the 1670s. Certainly the idea of limit, the idea that separates analysis from the rest of math, is fundamental to calculus. If you ever sat through a calculus class at school, you probably have some dim memory of a graph showing a curve with a straight line intersecting it at two points. “Now,” says the instructor, “if you let the two points come closer and closer together, in the limit…” and you forget the rest.

Calculus is not the whole of analysis—the divergence of the harmonic series is a theorem in analysis, but it does not belong to calculus, which did not exist in Nicole d’Oresme’s time. There are other quite large areas of analysis that do not strictly belong in calculus. Measure theory, for example, developed by Henri Lebesgue in 1901, and a chunk of set theory. I think it’s fair to say, though, that even these newer non-calculus areas of analysis were opened up with the idea of improving calculus—in Lebesgue’s case, of getting a better definition of “integral.”

The concepts that analysis deals with—“the infinite and the infinitesimal,” as Euler would have said; “limits and continuity,” his modern counterpart would insist—are among the most difficult for the human mind to grasp. This is why calculus is so fearsome to so many intelligent people. The causes of all the bafflement were stated very early on in the history of math—in about 450 B.C.E., by a Greek philosopher named Zeno. How (asked Zeno) is motion possible? How can we say that an arrow moves, if, at any given instant, it must be somewhere? If all time is composed of instants, and motion is not possible in any given instant, then how is motion possible at all?

In the early eighteenth century, when calculus first became known to the general educated public, the notion of infinitesimals came in for much scorn. The Irish philosopher George Berkeley (1685–1753—the California town is named after him) was a notable skeptic: “And what are these evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”

The difficulty people have in grasping these ideas is a reminder that mathematical thinking is, at some level, deeply unnatural. It goes against all the grain of human thought and language. Never mind analysis, this is true even of basic arithmetic. In the preface to Principia Mathematica, Whitehead and Russell note that

(They weren’t kidding. Principia Mathematica takes 345 pages to define the number “1.”)

This is surely right. A whale is, by any standard of complexity that makes sense, a vastly more complicated thing than “five,” yet it is a much easier thing for the human mind to apprehend. Any tribe of human beings that was acquainted with whales would certainly have a word for them in their language; yet there are peoples whose language has no word for “five” even though five-ness is there, quite literally, at their fingertips! I repeat, mathematical thinking is a deeply unnatural way of thinking, and this is probably why it repels so many people. And yet, if that repulsion can just be overcome, what benefits flow! Consider the 2,000-year struggle to domesticate the concept of “zero”—a number widely accepted as mathematically legitimate only about 400 years ago. Where should we be nowadays without it?

Arithmetic, by contrast with analysis, is widely taken to be the easiest, most accessible branch of math. Whole numbers? Obviously useful for counting. Negative numbers? Indispensable if you want to know the temperature on a cold day. Fractions? Well, of course I know that a nut won’t fit onto a bolt. If you gave me a little time with paper and pencil, I could probably tell you whether a nut could fit on a bolt. What’s to be afraid of?

In fact, arithmetic has the peculiar characteristic that it is rather easy to state problems in it that are ferociously difficult to prove. It was in 1742 that Christian Goldbach put forward his famous conjecture that every even number greater than 2 can be expressed as the sum of two primes. Twenty-six decades of effort by some of the best minds on the planet have failed to prove or disprove this simple assertion (which has inspired at least one novel, Apostolos Doxiadis’s Uncle Petros and Goldbach’s Conjecture²⁹). There are a thousand conjectures like this in arithmetic³⁰; some proved, most still open.

This is undoubtedly what Gauss had in mind when he declined to enter into a prize competition for the solution of Fermat’s Last Theorem. To Heinrich Olbers, who had urged him to compete, Gauss replied “I confess that Fermat’s Theorem … has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”

Gauss’s indifference is in this case a minority viewpoint, it must be said. A problem that can be stated in a few plain words, yet which defies proof by the best mathematical talents for decades or—in the case of Goldbach’s Conjecture or Fermat’s Last Theorem—for centuries, has an irresistible attraction for most mathematicians. They know that they can achieve great fame by solving it, as Andrew Wiles did when he proved Fermat’s Last Theorem. They know, too, from the history of their subject, that even failed attempts can generate powerful new results and techniques. And there is, of course, the Mallory factor. When the New York Times asked George Mallory why he wanted to climb Mount Everest, Mallory replied: “Because it’s there.”

V. The connection between measuring and continuity is this. Since there is no theoretical limit to the accuracy with which a quantity can be measured, the list of all possible measurements is infinite, and in-

finitely fine. Between a measurement of 2.3 inches and one of 2.4 inches there are intermediate, more precise, measurements of 2.31, 2.32, 2.33…, 2.39 inches; and these in turn can be subdivided ad infinitum. We can, therefore, in imagination, travel connectedly from any measuring number to any other, passing over the infinitude of other measuring numbers that lie between them, without ever finding ourselves without (so to speak) a number to stand on. This idea of connectedness—of traversing some space or some interval without ever having to leap over a void—lies behind the vitally important mathematical concepts of continuity and limit. In other words, it lies behind all of analysis.

When counting, by contrast, there is nothing between seven and eight; we must leap from one to the other, with no stepping-stones in between. You can measure something at seven and a half units, but you can’t count seven and a half objects. (You might object to this, saying “What if I say I have seven and a half apples? Isn’t that a counting statement?” To which the answer is “I might allow you to say so … but only if you’re sure that is precisely one-half of an apple, as precisely as Larry, Curly, and Moe are precisely three people. Could it not be 0.501 of an apple, or 0.497…?” And at once, if we want to resolve the issue, we must pass into the realm of measuring. “Seven and a half string quartets” is just cheating.)

The great fusion between arithmetic and analysis—between counting and measuring, between numbers staccato and numbers legato—came about as the result of an inquiry into prime numbers, conducted by Lejeune Dirichlet in the 1830s. Dirichlet (1805–1859) was, names notwithstanding, German, from a small town near Cologne, where he got most of his education.³¹ The fact that he was a German deserves a brief detour by itself; for the fusion of ideas from arithmetic and analysis, carried out by Dirichlet and Riemann, happened within a broader social change in mathematics at large, the rise of the Germans.

VI. If you draw up a list of the dozen or so greatest mathematicians at work in 1800, it looks something like this: Argand, Bolyai, Bolzano, Cauchy, Fourier, Gauss, Germain, Lagrange (just), Laplace, Legendre, Monge, Poisson, Wallace. A different writer, or this writer in a different mood, might of course add a name here or subtract one there, but without making any difference to the most striking feature of the list, which is the near-total absence of Germans. Gauss is the only one. There is one Scot, one Czech, one Hungarian, and one “disputed” (Lagrange, baptized Giuseppe Lagrangia, is claimed by both Italy and France). The rest are all French.

There were a great many more mathematicians at work in 1900, so a list made up for that year would be correspondingly more likely to start a fistfight. However, I believe that the following attains some local minimum of controversiality: Borel, Cantor, Carathéodory, Dedekind, Hadamard, Hardy, Hilbert, Klein, Lebesgue, Mittag-Leffler, Poincaré, Volterra. Four Frenchmen, an Italian, an Englishman, a Swede, and five Germans.³²

The rise of the Germans to prominence in mathematics is intimately related to some of the historical events I sketched in Chapters 2 and 4. For all of Frederick the Great’s reforms, the defeat at Jena in 1806 showed the Prussians that they still had some way to go in modernizing and strengthening their state. The rising nationalist passions stimulated by the long wars against Napoleon, and by the Romantic Movement, were an added spur to reform, in spite of having been thwarted (as the nationalists saw it) by the failure of the Congress of Vienna to unify the German-speaking peoples. In the years after Jena, the Prussian army was reorganized on a basis of universal conscription, serfdom was abolished, restrictions on industry were lifted, taxation and the whole financial system were overhauled, and the educational reforms of Wilhelm von Humboldt, already mentioned in Chapter 2.iv, were instituted. The lesser German states took their lead from Prussia, and Germany at large soon became a place hospitable to science, industry, progress, education—and, of course, mathematics.

It should perhaps be added that there was another, lesser, reason for the rise of nineteenth-century German mathematics. There was Gauss. His is the only German name in that list I drew up for 1800; but there go ten dimes to the dollar, and one Gauss was worth at least ten ordinary mathematicians. The fact that Gauss was in his observatory at Göttingen and teaching (though he disliked teaching and did as little as he could get away with) was sufficient to put Germany, and Göttingen, on the mental map of anyone interested in mathematics.

VII. That is the world in which Lejeune Dirichlet grew up. Born in 1805, he was of the generation before Riemann. The son of a postmaster in a small town 20 miles east of Cologne, in Prussia’s Rhine province, Dirichlet was also among the first generation to benefit from von Humboldt’s reformed gymnasium system of secondary education. He must have been an exceptionally quick study, for by age 16 he had acquired all the qualifications necessary for university entrance. Already hooked on mathematics, he set off for what was still the world capital of mathematical knowledge, Paris, carrying with him the book he treasured above all others, Gauss’s Disquisitiones Arithmeticae. In Paris, 1822–1825, Dirichlet attended lectures given by many of the great French stars of that time, including at least four from the list I presented earlier: Fourier, Laplace, Legendre, and Poisson.

In 1827, now 22 years old, Dirichlet returned to Germany to teach at the University of Breslau in Silesia. (Breslau is now in Poland, and appears on modern maps as the city of Wrocław.) He gained this position with the assistance and encouragement of Alexander von Humboldt, the explorer, and brother of Wilhelm. Both von Humboldts were key players in these early nineteenth-century German cultural developments.

Outside Berlin, however, German universities were in the condition I have described in Chapter 2.vii, given over mainly to the training of teachers, lawyers, and so on. Dissatisfied with Breslau, Dirichlet got a position in Berlin and spent most of his professional career— 1828–1855—teaching there. Among those he taught was a brilliant but shy young scholar from the Wendland region of north Germany, Bernhard Riemann, who had transferred from the University of Göttingen in search of the finest mathematical instruction. I shall have much more to say about Dirichlet’s influence on Riemann in Chapter 8; here I note only the connection, and the fact that through it, Riemann came to revere Dirichlet, considering him to be the second greatest mathematician alive, after Gauss.

Dirichlet married Rebecca Mendelssohn, one of the sisters of the composer Felix Mendelssohn, thereby forming one of the many Mendelssohn-mathematics connections.³³

We have some sketches of Dirichlet and his teaching style during his Berlin years from Thomas Hirst, an English mathematician and diarist who spent much of the 1850s traveling in Europe, taking in mathematics wherever he could find it. During the fall and winter of 1852–1853 he was in Berlin, where he befriended Dirichlet and attended his lectures. From Hirst’s diary:

VIII. For the purposes of this story, Dirichlet’s principal significance is as follows. Inspired by a result Euler had proved precisely 100 years before, a result I hereby name “the Golden Key,” Dirichlet in 1837 brought together ideas from analysis and arithmetic to prove an important theorem about prime numbers. This is generally considered to be the beginning of analytic number theory; of arithmetic with limits. The title of Dirichlet’s groundbreaking paper was, I am sorry to say, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält—“Proof of the theorem that each unlimited arithmetic progression, whose first member and difference are whole numbers without common factor, contains infinitely many prime numbers.”

Take any two positive whole numbers and repetitively add one to the other. If the two numbers have a common factor, every resulting number has that factor, too; repetitively adding 6 to 15 gives you 15, 21, 27, 33, 39, 45, … all of which have 3 as a factor. If the two numbers have no common factor, however, there is the possibility of getting some primes in the list. If, for example, I repetitively add 6 to 35, I get 35, 41, 47, 53, 59, 65, 71, 77, 83, … which has lots of primes—along, of course, with many non-primes like 65 and 77. How many primes? Could this sequence contain an infinity of primes? In other words, could it be that, for any number N, no matter how big, I could, by

repetitively adding 6 to 35 enough times, turn up more than N primes? Could any sequence like this, made from any two numbers with no common factor, contain an infinity of primes?

Yes, it could. This is, in fact, precisely the case. Take any two numbers with no common factor and repetitively add one to the other. You will generate an infinity of primes (mixed with an infinity of non-primes). Gauss had conjectured that this was the case—knowing Gauss’s powers, one is tempted to say that he intuited it—but it was decisively proved by Dirichlet in that 1837 paper. It was in Dirichlet’s proof that the first part of the great fusion was accomplished.

The truth is even more interesting. Take any positive whole number, say, 9. How many of the numbers less than 9 have no factor in common with it, not counting 1 as a factor? Well, there are six such numbers, and here they are: 1, 2, 4, 5, 7, 8. Take each one of these in turn, and repetitively add 9 to it.

1: 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, …

2: 11, 20, 29, 38, 47, 56, 65, 74, 83, 92, 101, 110, 119, 128, …

4: 13, 22, 31, 40, 49, 58, 67, 76, 85, 94, 103, 112, 121, 130, …

5: 14, 23, 32, 41, 50, 59, 68, 77, 86, 85, 104, 113, 122, 131, …

7: 16, 25, 34, 43, 52, 61, 70, 79, 88, 97, 106, 115, 124, 133, …

8: 17, 26, 35, 44, 53, 62, 71, 80, 89, 98, 107, 116, 125, 134, …

Not only does every one of those sequences contain an infinity of primes (I have underlined them), but each of the six sequences contains the same proportion of primes. In other words, if you imagine each sequence stretching out to the neighborhood of some very large number N, instead of merely to the neighborhood of 134, then each contains about the same number of primes, about , if the Prime Number Theorem is true (which had not yet been proved in Dirichlet’s time). If N is 134, is about 4.55983336…. The six sequences I’ve shown turn up 5, 5, 4, 5, 4, and 5 primes, for an

average of 4.6666…; high by 2.3 percent, which is pretty good for such a small sample size.

To prove his result, Dirichlet began with a form of arithmetic developed at great length by Gauss in Disquisitiones Arithmeticae. Mathematicians call it “the arithmetic of congruences.” You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, … up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 ≡ 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.” The phrase “modulo twelve” means “I am working from a clock-face with twelve hours marked, 0 to 11.” This may seem trivial, but in fact the arithmetic of congruences goes very deep and is full of strange and difficult results. Gauss was a great grand master of it; not one of the seven sections of Disquisitiones Arithmeticae is free from that “≡” sign.

The Disquisitiones, remember, was the constant companion of Dirichlet’s younger years. When he came to this problem, in 1836 or 1837, he was in his early 30s and must have completely internalized Gauss’s work on congruences. Then somehow, Euler’s 1737 result— “the Golden Key”—came to his attention. It gave him an idea; he put the two things together, applied some elementary techniques of analysis, and got his proof.

IX. Dirichlet was thus the first to pick up the Golden Key, the link between arithmetic and analysis, and make serious use of it. In terms of the analogy I am using, it would be a bit too much to say that he turned the key. I would rather say that he picked it up, sensed its beauty and potential power, set it down again, then used it as a model for a similar key—a silver key, you might say—to unlock the particular problem he had in front of him. The great fusion, analytic number

theory, did not appear in its full glory until 22 years later, in Riemann’s paper of 1859.

Recall, though, that Riemann was one of Dirichlet’s students and certainly knew of the older man’s work. In the opening paragraph of the 1859 paper, in fact, he mentioned Dirichlet’s name in conjunction with that of Gauss. They were his two mathematical idols. If it was Riemann who turned the key, it was Dirichlet who first showed it to him and demonstrated that it was a key to something or other; and it is to Dirichlet that the immortal glory of inventing analytic number theory properly belongs.

But what, exactly, is this Golden Key? What was it that Leonhard Euler, working away by candlelight in his room, the secret police of the Bironovschina prowling the streets of St. Petersburg outside, left lying around for Dirichlet to find a hundred years later?