What is Riemann’s Hypothesis?

Barry Mazur William Stein

June 28, 2007

Contents

1 Foreword 2

2 Thoughts about numbers: ancient, medieval, and modern 4

3 What are prime numbers? 5

4 Questions about primes that any person might ask 8

5 How many primes are there? 9

6 Prime numbers viewed from a distance 14

7 “Pure” and “Applied” Mathematics 15

8 A probabilistic “first” guess 16

9 What is a “good approximation”? 17

10 What is Riemann’s Hypothesis? (second formulation) 18

11 The information contained in the staircase of primes 20

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12 Tinkering with the carpentry of the staircase of primes. 20

13 To our readers of Part I. 23

14 How Calculus manages to find the slopes of graphs that have no slopes 24

15 What do primes, heat, and sound have in common? 30

16 Spike distributions that have discrete spectrum 31

17 The Spike distribution Φ(t); the Riemann Hypothesis (fourth formulation) 31

18 The undistorted staircase: returning to P (N) 34

19 William: some alternative text about Riemann 43

20 Further alternative text here 44

21 The Riemann Zeta-Function; and Riemann’s Hypothesis (fifth version) 45

22 Some fragments to be either used or discarded 46

23 Glossary 47

1 Foreword

The Riemann Hypothesis is one of the great unsolved problems of mathematics and the reward of$1,000,000 of Clay Mathematics Institute prize money awaits the person who solves it. But—withor without money—its resolution is crucial for our understanding of the nature of numbers.

There are at least four full-length books recently published, written for a general audience, thathave the Riemann Hypothesis as their main topic. A reader of these books will get a fairly richpicture of the personalities engaged in the pursuit, and of related mathematical and historicalissues.

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This is not the mission of the booklet that you now hold in your hands. We aim—instead—toexplain, in as direct a manner as possible and with the least mathematical background required,what this problem is all about and why it is so important. For even before anyone proves thishypothesis to be true (or false!), just getting familiar with it and with some of the the ideas behindit, is exciting. Moreover, this hypothesis is of crucial importance in a wide range of computationalmathematics, even if it is never proved, since it gives us an excellent sense of how long certaincomputer programs will take to run, which bolsters our confidence to initiate a computation thatmight take weeks or even months to complete.

Our “booklet” comes in two pieces: the actual text which is a mere *** pages, and a CD which hasa gallery of illustrations that are meant to accompany the text. There are very few mathematicalequations in our text. We believe that a complete reading of our booklet and viewing of theaccompanying illustrations need not take more than *** hours. But for the more adventurousreaders, we have included in the CD something we call worksheets that explain to the reader howto experiment with the parameters for the ranges of data illustrated, so as to get a vivid sense ofhow the numbers ‘behave.’

So, what is the Riemann Hypothesis? Here is a first formulation of this hypothesis, put in oneridiculously condensed paragraph. We will be giving other, quite miraculous, formulations of thishypothesis as well, but the task of our booklet is to expand the following paragraph into a realexplanation and to convince you of its importance and its beauty.

What sort of Hypothesis is the Riemann Hypothesis?

It has long been known that that for any real number X the number of prime numbersless than X (denoted π(X)) is approximately X/ logX in the sense that the ratio π(X)

X/ log Xtends to 1 as X goes to infinity. The Riemann Hypothesis would give us a much moreaccurate “count” for π(X) in that it will offer a specific smooth function R(X) (hardly anymore difficult to describe than X/ logX) and then conjecture that R(X) is an essentiallysquare root accurate approximation to π(X); i.e., for any given exponent greater than1/2 (you choose it: 0.501, 0.5001, 0.50001, for example) and for large enough X wherethe phrase “large enough” depends on your choice of exponent the error term i.e., thedifference between R(X) and the number of primes less than X in absolute value is lessthan X raised to that exponent (e.g. < X0.501, < X0.5001, etc.)

In this paragraph we have put in boldface three key phrases, prime numbers, square-rootaccurate, and error term. A famous mathematician, Raoul Bott, once said—giving advice tosome young mathematicians—that whenever one reads a mathematics book or article, or goes toa math lecture, one should aim to come home with something very specific (it can be small, butshould be specific) that has application to a wider class of mathematical problem than was thefocus of the text or lecture. If we were to suggest some possible specific items to come home with,after read our booklet, our three key- phrases would head the list. As for words of encouragementto think hard about the first of these, i.e., prime numbers, we can do no better than to quote aparagraph of Don Zagier’s classic 12-page exposition of all this mathematics, The First 50 MillionPrime Numbers:

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“There are two facts about the distribution of prime numbers of which I hope to convince you sooverwhelmingly that they will be permanently engraved in your hearts. The first is that, [they are]the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among thenatural numbers, seeming to obey no other law than that of chance, and nobody can predict wherethe next one will sprout. The second fact is even more astonishing, for it states just the opposite:that the prime numbers exhibit stunning regularity, that there are laws governing their behavior,and that they obey these laws with almost military precision.”1

2 Thoughts about numbers: ancient, medieval, and modern

If we are to believe Aristotle, the early Pythagoreans thought that the principles governing Numberare “the principles of all things,” the elements of number being more basic than the Empedocleanphysical elements earth, air, fire, water. To think about number is to get close to the architectureof “ what is.”

So, how far along are we in our thoughts about numbers?

Rene Descartes, almost four centuries ago, expressed the hope that there soon would be “almostnothing more to discover in geometry.” Contemporary physicists dream of a “final theory.” Butdespite its venerability and its great power and beauty, the pure mathematics of numbers may stillbe in the infancy of its development, with depths to be explored as endless as the human soul, andnever a final theory.

Figure 2.1: Don Quixote and “his” Dulcinea del Toboso

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Numbers are obstreperous things. Don Quixote encountered this when he requested that the“bachelor” compose a poem to his lady Dulcinea del Toboso, the first letters of each line spellingout her name. The “bachelor” found

“a great difficulty in their composition because the number of letters in her name was17, and if he made four Castilian stanzas of four octosyllabic lines each, there would beone letter too many, and if he made the stanzas of five octosyllabic lines each, the onescalled decimas or redondillas, there would be three letters too few...”

”It must fit in, however, you do it,” pleaded Quixote, not willing to grant the imperviousness ofthe number 17 to division.

Seventeen is indeed a prime number: there is no way of factoring it as the product of smallernumbers, and this accounts—people tell us—for its occurrence in some phenomena of nature, aswhen last year the 17-year cicadas all emerged to celebrate a “reunion” of some sort in our fieldsand valleys.

Prime numbers, despite their primary position in our modern understanding of number, were notspecifically doted over in the ancient literature before Euclid, at least not in the literature thathas been preserved. Primes are mentioned as a class of numbers in the writings of Philolaus(a predecessor of Plato); they are not mentioned specifically in the Platonic dialogues, whichis surprising to me given the intense interest Plato had in mathematical developments; and theymake an occasional appearance in the writings of Aristotle, which is not surprising, given Aristotle’semphasis on the distinction between the composite and the incomposite. “The incomposite is priorto the composite,” writes Aristotle in Book 13 of the Metaphysics.

But, until Euclid, prime numbers seem not to have been singled out as the extraordinary math-ematical concept, central to any deep understanding of numerical phenomena, that they are nowunderstood to be.

There is an extraordinary wealth of established truths about numbers; these truths provoke sheerawe for the beautiful complexity of prime numbers. But each of the important new discoveries wemake give rise to a further richness of questions, educated guesses, heuristics, expectations, andunsolved problems.

3 What are prime numbers?

Primes as atoms. To begin from the beginning, think of the operation of multiplication as abond that ties numbers together: the equation 2 × 3 = 6 invites us to imagine the number 6 as

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(a molecule, if you wish) built out of its smaller constituents 2 and 3. Reversing the procedure,if we start with a whole number, say 6 again, we may try to factor it (that is, express it as aproduct of smaller whole numbers) and, of course, we would eventually, if not immediately, comeup with 6 = 2× 3 and discover that 2 and 3 factor no further; the numbers 2 and 3, then, are theindecomposable entities (atoms, if you wish) that comprise our number. By definition, a primenumber (colloquially, a prime) is a whole number, bigger than 1, that cannot be factored into aproduct of two smaller whole numbers. So, 2 and 3 are the first two prime numbers. The nextnumber along the line, 4, is not prime, for 4 = 2 × 2; the number after that, 5, is. Primes are,multiplicatively speaking, the building blocks from which all numbers can be made. A fundamentaltheorem of arithmetic tells us that any number (bigger than 1) can be factored as a product ofprimes, and the factorization is unique except for rearranging the order of the primes. For example,if you try to factor the number 300, there are many ways to begin:

300 = 30× 10 or 300 = 6× 50

and there are various other starting possibilities. But if you continue the factorization (“climbingdown” any one of the possible “factoring trees”) to the bottom, where every factor is a primenumber, e.g., in Figure 3.1, you always end up with the same collection of prime numbers:

300 = 22 × 3× 52.

300

25 12

5 5 3 4

2 2

300

15 20

5 3 4 5

2 2

Figure 3.1: A factor tree that illustrates the factorization of 300 as a product of primes.

646969323016422 393965

322 51 11 35815161 2 17 3 7163 5

7 23 29 24713 19

Figure 3.2: Factorization tree for 6469693230

The Riemann Hypothesis probes the question: how intimately can we know prime numbers, thoseatoms of multiplication? Anytime we visit a website and purchase something online, prime numbers

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having over 150 decimal digits are used to keep our bank transactions private. This ubiquitous useto which these giant primes are put depends upon a very simple principle: it is much easier tomultiply numbers together than to factor them. If you had to factor, say, the number 143 youmight scratch your head for a few minutes before discovering that 143 is 11 × 13. But if you hadto multiply 11 by 13 you would do it straightaway. Offer two primes, say, P and Q each with morethan 100 digits, to your computing machine and ask it to multiply them together: you will gettheir product N = P ×Q with its 200 or so digits in nanoseconds. But present that number N toany current desktop computer, and ask it to factor N– and the computer will fail to do the task.The safety of much encryption depends upon this guaranteed1 failure!

If we were latter-day number-phenomenologists we might revel in the discovery and proof that

p = 232582657 − 1

is a prime number, this number having 9,808,358 digits! There are, in fact, infinitely many primenumbers, as was proved in Euclid’s Elements as follows. Suppose there are only finitely manyprimes p1, . . . , pn. Let n = p1p2 · · · pn + 1. Then n is divisible by some prime, but no pi divides n,which is contrary to our assumption that p1, . . . , pn is the complete list of primes.

The number p displayed above is the largest prime we know, where by “know” we mean that weknow it so explicitly that we can compute things about it. For example, the last two digits of p are71. Of course p is not the largest prime number since there are infinitely many primes, e.g., thenext prime q after p is a prime. But there is no obvious way to compute anything about q. Forexample, what is the last digit of q in its decimal expansion?

Eratosthenes, the mathematician from Cyrene (and later, librarian at Alexandria) explained howto sift the prime numbers from the series of all numbers: in the sequence of numbers,

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29,

for example, start by circling the 2 and crossing out all the other multiples of 2. Next, go backto the beginning of our sequence of numbers and circle the first number that is neither circled norcrossed out (that would be, of course, the 3); then cross out all the other multiples of 3. This givesthe pattern: go back again to the beginning of our sequence of numbers and circle the first numberthat is neither circled nor crossed out; then cross out all of its other multiples. Repeat this patternuntil all the numbers in our sequence are either circled, or crossed out, the circled ones being theprimes.

In Figure ?? we use the primes 2, 3, 5, and 7 to sieve out the primes up to 100.

Especially if you have had little experience with math, may I suggest that you actually followEratosthenes’ lead, and perform the repeated circling and crossing-out to watch the primes emerge,intriguingly staggered through our sequence of numbers,

2 3 • 5 • 7 • • • 11 • 13 • • • 17 • 19 • • • 23 • • • • • 29, . . .

1Nobody has ever proved that there is no fast way to factor integers.

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4 Questions about primes that any person might ask

We become quickly stymied when we ask quite elementary questions about the spacing of theinfinite series of prime numbers.

For example, are there infinitely many pairs of primes whose difference is 2? The sequence on thepage seems to be rich in such pairs

5− 3 = 2, 7− 5 = 2, 13− 11 = 2, 19− 17 = 2,

and we know loads more such pairs (for example, there are 1,177,209,242,304 such pairs less than1,000,000,000,000,000) but the answer to our question, are there infinitely many?, is not known.Are there infinitely many pairs of primes whose difference is 4? Answer: equally unknown. Is everyeven number greater than 2 a sum of two primes? Answer: unknown. Are there infinitely manyprimes which are 1 more than a perfect square? Answer: unknown.

Is there some neat formula giving the next prime? More specifically, If I give you a number N ,say N = one million, and ask you for the first number after N that is prime, is there a methodthat answers that question without, in some form or other, running through each of the successivenumbers after N rejecting the nonprimes until the first prime is encountered? Answer: unknown.

Here is a curious question that you can easily begin to check out for small numbers. We know, ofcourse, that the even numbers and the odd numbers are nicely and simply distributed: after everyodd number comes an even number, after every even, an odd, there is an equal number of oddnumber as even numbers less than any given odd number; and there may be nothing else of interestto say about the matter. Things change considerably, though, if we focus our concentration onmultiplicatively even numbers and multiplicatively odd numbers.

A multiplicatively even number is one that can be expressed as a product of an even number ofprimes; and a multiplicatively odd number is one that can be expressed as a product of an oddnumber of primes. So, any prime is multiplicatively odd, the number 4 = 2 · 2 is multiplicativelyeven, and so is 6 = 2 · 3, 9 = 3 · 3, and 10 = 2 · 5; but 12 = 2 · 2 · 4 is multiplicatively odd. Here issome data:

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Table 4.1: Multiplicatively even and odd numbers ≤ N

N 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Multiplicatively even 1 1 2 2 3 3 3 4 5 5 5 5 6 7 8Multiplicatively odd 1 2 2 3 3 4 5 5 5 6 7 8 8 8 8

Now looking at this data, a natural, and simple, question to ask about the concept of multiplicativeoddness and evenness is:

Is there some N ≥ 2 for which there are more multiplicatively even numbers less than or equal toN than multiplicatively odd ones?

A negative response to this question implies the Riemann Hypothesis. In contrast to the list ofprevious questions, the answer to this one is known: alas, there is such an N . In 1960, Lehmanshowed that for N = 906, 400, 000 there are 708 more multiplicatively even numbers up to Nthan multiplicatively odd numbers (Tanaka found in 1980 that the smallest N with this propertyis N = 906, 150, 257). For more details, see P. Borwein, Sign changes in sums of the LiouvilleFunction.

WILLIAM: Do you think it might be worth putting in a graph for this data? I would have said thatthis is our “second” formulation of RH except for the curous fact that if one can prove the above,that would imply both RH and the fact that the zeroes are all simple. For a reference here, there is anice short paper of Norbert Wiener Notes on Polya’s and Turan’s hypothesis concerning Liouville’sfactor on this (page 765 of volume II of Wiener’s Collected Works; also: G. Polya VershiedeneBemerkungen zur Zahlentheorie Jahresbericht de Deutschen Mathematiker-Vereinigung, 28 (1919)31-40

These are questions that have been asked about primes (and we could give bushels more), questionsexpressible in simple vocabulary, that we can’t answer today. We have been studying numbers forover two millenia and yet we are indeed in the infancy of our understanding.

5 How many primes are there?

Slow as we are to understand primes, at the very least we can try to count them. You can see thatthere are 10 primes less than 30, so you might encapsulate this by saying that the chances that anumber less than 30 is prime is 1 in 3. This frequency does not persist, though; here is some moredata: There are 168 primes less than a thousand (so we might say that among the numbers lessthan a thousand the chances that one of them is prime is roughly 1 in 6).

There are 78, 498 primes less than a million (so we might say that the chances that a random choiceamong the first million numbers is prime have dropped to roughly 1 in 13).

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50 100 150 200

0.2

0.4

0.6

Figure 5.1: Proportion of primes up to N = 200

10

250 500 750 1000

0.2

0.4

0.6

Figure 5.2: Proportion of primes up to N = 1100

11

2500 5000 7500 10000

0.2

0.4

0.6

Figure 5.3: Proportion of primes up to N = 11000

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There are 455, 052, 512 primes less than ten billion; i.e., 10,000,000,000 (so we might say that thechances are down to roughly 1 in 22).

Primes, then, seem to be thinning out. Return to the sifting process we did earlier, and take a lookat a few graphs, to get a sense of why that might be so. There are a hundred numbers less than orequal to 100, a thousand numbers less than or equal to 1000, etc.: the shaded bar graph that lookslike a regular staircase, each step the same length as each riser, climbing up at, so to speak, a 45degree angle, counts all numbers up to and including N .

Following Eratosthenes, we have sifted those numbers, to pan for primes. Our first move was tothrow out roughly half the numbers (the even ones!) after the number 2. The cross-hatched bargraph in this figure which is, with one hiccup, a regular staircase climbing at a smaller angle, eachstep twice the lengths of each riser, illustrates the numbers that are left after one pass throughEratosthenes’ sieve, which includes, of course, all the primes. So, the chances that a number biggerthan 2 is prime is at most 1 in 2. Our second move was to throw out a good bunch of numbersbigger than 3. So, the chances that a number bigger than 3 is prime is going to be even less. Andso it goes: with each Eratosthenian move in our sieving process we are winnowing the field moreextensively, reducing the chances that the later numbers are prime.

All Numbers

Sieve by 2

Primes

50 100 150 200

50

100

150

200

All Numbers

Primes

250 500 750 1000

250

500

750

1000

Figure 5.4: Sieving by removing multiples of 2 up to 100 and Sieving for primes up to 1000

The red curve in these figures actually counts the primes: it is the beguilingly irregular staircaseof primes. Its height above any number N on the horizontal line records the number of primesless than or equal to N , the accumulation of primes up to N . Refer to this number as P (N). SoP (2) = 1, P (3) = 2, P (30) = 10; of course, if you believed some of the data above you could plota few more values of P (N), like P (ten billion) = 455, 052, 512.

Let us accompany Eratosthenes for a few further steps in his sieving process. Here is a graph ofall whole numbers after we have removed the even numbers greater than 2, and the multiples of 3greater than 3 itself:

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From this graph you can see that if you go “out a way” the likelihood that a number is a prime isless that 1 in 3. Here is what Eratosthenes sieve looks like after sifting 2, 3, 5, and 7,

This data may begin to suggest to you that as you go further and further out on the number linethe percentage of prime numbers among all whole numbers tends towards 0% (it does).

To get a sense of how the primes accumulate, we will take a look at the staircase of primes forN = 25, and N = 100.

P(N)

5 10 15 20 25

2

4

6

8 P(N)

25 50 75 100

5

10

15

20

25

Figure 5.5:

6 Prime numbers viewed from a distance

The striking thing about these figures is that as the numbers get large enough, the jagged accumu-lation of primes, those quintessentially discrete entities, becomes smoother and smoother, to theeye. How strange and wonderful to watch, as our viewpoint zooms out to larger ranges of numbers,the accumulation of primes taking on such a smooth, elegant, shape.

P(N)

250 500 750 1000

25

50

75

100

125

150

175

Figure 6.1:

But don’t be fooled by the seemingly smooth shape of the curve in the last figure above: it is justas faithful a reproduction of the staircase of primes as the typographer’s art can render, for thereare about 8, 000 tiny steps and risers in this curve, all hidden by the thickness of the print of thedrawn curve in the figure. It is already something of a miracle that we can approximately describe

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P(N)

2500 5000 7500 10000

250

500

750

1000

1250

Figure 6.2:

P(N)

25000 50000 75000 100000

2500

5000

7500

10000

Figure 6.3:

the build-up of primes, somehow, using a smooth curve. But what smooth curve?

That last question is not rhetorical. If I draw a curve with chalk on the blackboard, this can signifya myriad of smooth (mathematical) curves all encompassed within the thickness of the chalk-line,all–if you wish–reasonable approximations of one another. So, there are many smooth curves thatfit the chalk-curve. With this warning, but very much fortified by the data of figure 2b, let us ask:what is a smooth curve that is a reasonable approximation to the staircase of primes?

7 “Pure” and “Applied” Mathematics

At this point, we hope that we don’t wear down your patience, for we would like to indulge in adigression about the role of question- asking (as above) in mathematics. Loosely speaking, thereare two types of mathematics: pure and applied. Now, we probably put more theoretical math inthe applied category than most of our colleagues would. This is because we want a given piece ofmathematics to be categorized as “pure” or “applied” turning on whether the intention behind it,is that it eventually be used elsewhere (either within or outside the discipline of Mathematics)—inwhich case we’ll call it applied—or it is to explain and get at the root of some concept—in which casewe’ll call it pure. Of course, there is a great overlap. Moreover, many questions in mathematics are“hustlers” in the sense that, at first view, what is being requested is that some simple task be done(e.g., to find a smooth curve that is a reasonable approximation to the staircase of primes)—once

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successfully done, it will, presumably, have many useful by-products—that is, it will be a piece ofmathematics that we might classify as applied. But, the fuller motivation behind the question ispure: to strike behind the mask of the phenomenology of the mathematical situation, and get atthe hidden fundamentals that actually govern the phenomena.

The particular issue before us is, in our opinion, twofold, both applied, and pure: can we curve-fitthe “staircase of primes” by a well approximating smooth curve? The story behind this alone ismarvelous, has a cornucopia of applications, and we will be telling it below. But our curiosityhere is driven by a question that is pure, and less amenable to precise formulation: are theremathematical concepts at the root of, and more basic than (and “prior to,” to borrow Aristotle’suse of the phrase,) prime numbers–concepts that account for the apparent complexity of the natureof primes?

[[TODO (william): Insert a discussion here about how believing RH is crucial to doing manyalgebraic number theory calculations. E.g., complexity analyses for algorithms like factoring dependon RH. Also, standard class group algorithms are vastly faster if we assume RH (or even more!), sonumber theorists do so all the time when doing numerical experiments (though of course they try toremove the hypothesis...). See http://pari.math.u-bordeaux.fr/dochtml/html.stable/, e.g.,for some sense of what this is like in the trenches... It says “Warning. Make sure you understandthe above!”]]

8 A probabilistic “first” guess

The search for such approximating curves began, in fact, two centuries ago when Carl FriedrichGauss defined a certain beautiful curve that, experimentally, seemed to be an exceptionally goodfit for the staircase of primes. Let us denote Gauss’s curve G(X); it has an elegant simple formulacomprehensible to anyone who has had a tiny bit of calculus [If you make believe that the chancesthat a number N is a prime is inversely proportional to the number of digits of N you might wellhit upon Gauss’s curve]. In a letter written in 1849 Gauss claimed that as early as 1792 or 1793 hehad already observed that the density of prime numbers over intervals of numbers of a given roughmagnitude X seemed to average 1/logX.

Gauss was an inveterate computer: he wrote in his 1849 letter that there are 216, 745 prime numbersless than three million (This is wrong: the actual number of these primes is 216816). Gauss’s curvepredicted that there would be 216, 971 primes– a miss, Gauss thought, by 226 (but actually he wascloser than he thought: the correct miss is a mere 161; not as close as the recent US elections, butpretty close nevertheless). So you shouldn’t be too surprised if Figures 4 and 2b [[fix references]]look the same! Gauss’s computation brings up two queries: will this spectacular ”good fit” continuefor arbitrarily large numbers? and, the (evidently prior) question: what counts as a good fit?

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Area ~ 29.080978

25 Primes

1/log(x)25 50 75 100

-0.4

-0.2

0.2

0.4

0.6

0.8 Area ~ 176.564494

168 Primes

1/log(x)250 500 750 1000

-0.4

-0.2

0.2

0.4

0.6

0.8

Figure 8.1: The expected tally of the number of primes < X is equal to the area underneath thegraph from 1 to X.

9 What is a “good approximation”?

If you are trying to estimate a number, say, around ten thousand, and you get it right to withina hundred, let us celebrate this kind of accuracy by saying that you have made an approximationwith square-root error (

√10,000 = 100). Of course, we should really use the more clumsy phrase

“an approximation with at worst square-root error.” Sometimes we’ll simply refer to such approx-imations as good approximations. If you are trying to estimate a number in the millions, and youget it right to within a thousand, let’s agree that—again—you have made an approximation withsquare-root error (

√1,000,000 = 1,000). Again, for short, call this a good approximation. So, when

Gauss thought his curve missed by 226 in estimating the number of primes less than three million,it was well within the margin we have given for a “good approximation.”

More generally, if you are trying to estimate a number that has D digits and you get it almost right,but with an error that has no more than, roughly, half that many digits, let us say, again, that youhave made an approximation with square-root error or synomymously, a good approximation.

This rough account almost suffices for what we will be discussing below, but to be more precise,the specific gauge of accuracy that will be important to us is not for a mere single estimate of asingle error term,

Error term = Exact Value - Our “good approximation”

but rather for infinite sequences of estimates of error terms. Generally, if you are interested anumerical quantity q(X) that depends on the real number parameter X (e.g., q(X) could be π(X),“the number of primes < X”) and if you have an explicit candidate “approximation,” qapprox(X), tothis quantity let us say that qapprox(X) is essentially a square-root accurate approximationto q(X) if for any given exponent greater than 1/2 (you choose it: 0.501, 0.5001, 0.50001, . . . forexample) and for large enough X— where the phrase “large enough” depends on your choice ofexponent—the error term—i.e., the difference between qapprox(X) and the true quantity, q(X),is, in absolute value, less than qapprox(X) raised to that exponent (e.g. < X0.501, < X0.5001, etc.).

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Readers who know calculus and wish to have a technical formulation of this definition of goodapproximation might turn to the endnote 2 for a precise statement.

Remark 9.1. To get a feel for how basic the notion of approximation to data being square rootclose to the true values of the data is—and how it represents the “gold standard” of accuracy forapproximations, consider this fable.

Imagine that the devil had the idea of saddling a large committee of people with the task of findingvalues of P (N) for various large numbers N . This he did in the following somewhat ridiculousmanner, having already worked out which numbers are prime and which composite, himself. Sincethe devil is, as everyone knows, into the details, he has made no mistakes: his work is entirelycorrect. He gives each committee member a copy of the list of all whole numbers between 1 andone of the large numbers N in which he was interested, the devil having put check-marks by thenumbers on that list that are prime numbers, and no mark by the composite numbers. Now eachcommittee member would count the number of primes by considering each number, in turn, ontheir list to figure out whether or not there is a check beside it, and tally up the ones that havechecks. But since they are human, they will indeed be making mistakes, say 0.001% of the time.Assume further that it is just as likely for them to make the mistake of thinking that a numberhas a check-mark that does not have one, as the contrary: that a number hasn’t got a mark,when it does. If many people are engaged in such pursuit, some of them might over-count P (N);some of them might under-count it. The average error (over-counted or undercounted) would beproportional to

√N .

10 What is Riemann’s Hypothesis? (second formulation)

Recall that in our first formulation, the Riemann Hypothesis approximated π(X), the number ofprimes < X, by the function X/ logX, while Gauss’s guess was in terms of the area in the regionfrom 2 to X under the graph 1/ logX, a quantity sometimes referred to as Li(X) (pronounced Li-of-X)2. The two functions X/ logX and Li(X) are relatively close: they are essentially square-rootaccurate approximations one of the other. Therefore it is easy to see that X/ logX is an essentiallysquare-root accurate approximation to π(X) if and only if Li(X) is such. Here is a graph of thethree functions X/ logX, Li(X), and π(X) in the range ***.

William: CAN WE PUT A GRAPH OF THIS HERE?

Note that Li(X) is a “closer fit” to π(X) (at least in this range) so it is natural to give this secondformulation of Riemann’s Hypothesis—perfectly equivalent to the first formulation that we madein section 1:

2“Li” is short for logarithmic integral,R X

2dx/ log x.

18

The Riemann Hypothesis (second formulation)

For any real number X the number of prime numbers less than X is approximately Li(X)and that this approximation is essentially square root accurate.

Even though Riemann’s Hypothesis has not yet been proven (or disproven) we know that bothfunctions X/logX (as in our “first formulation”) and Li(X) (as in this “second formulation”) arerespectably close approximations to π(X). We know, for example, that for any number greaterthan 1 that you give (we will illustrate this by imagining that you gave the number 1.000000001) ifyou take X sufficiently large (and how large “sufficiently large” is depends on the number that yougave) π(X) is no greater than 1.000000001 times Li(X), and Li(X) is no greater than 1.000000001times π(X). The same is true if Li(X) is replaced by X/ logX. These facts are known as The PrimeNumber Theorem, a very hard-won piece of mathematics! (See the Appendix for a discussion ofthis theorem.)

The elusive Riemann Hypothesis, however, is much deeper than the Prime Number Theorem,and takes its origin from some awe-inspiring, difficult to interpret, lines in Bernhard Riemann’smagnificent 8-page paper, “On the number of primes less than a given magnitude,” published in1859.

Figure 10.1: From Riemann’s Manuscript

PUT FIGURES K,L HERE;– these are some of the jpg files of Riemann’s manuscriptthat Jim Carleson sent to me.

Riemann’s hypothesis, as it is currently interpreted, turns up as relevant, as a key, again and againin different parts of the subject: if you accept it as hypothesis you have an immensely powerfultool at your disposal: a mathematical magnifying glass that sharpens our focus on number theory.But it also has a wonderful protean quality– there are many ways of formulating it, any of theseformulations being provably equivalent to any of the others.

This Riemann Hypothesis remains unproved to this day, and therefore is “only a hypothesis,” asOsiander said of Copernicus’s theory, but one for which we have overwhelming theoretical andnumerical evidence in its support. It is the kind of conjecture that Frans Oort might label asuffusing conjecture in that it has unusually broad implications: many many results are now knownto follow, if the conjecture, familiarly known as RH, is true. A proof of RH would, therefore, fallinto the applied category, given our discussion above. But however you classify RH, it is a centralconcern in mathematics to find its proof (or, a counter-example!).

19

11 The information contained in the staircase of primes

We have borrowed the phrase “staircase of primes” from the popular book The Music of Primes byMarcus du Sautoi, for we feel that it captures the sense that there is a deeply hidden architectureto the graphs that compile the number of primes (up to N) and also because—in a bit—we will betinkering with this carpentry. Before we do so, though, let us review what this staircase looks like,for different ranges.

P(N)

5 10 15 20 25

2

4

6

8

Figure 11.1:

P(N)

25 50 75 100

5

10

15

20

25

Figure 11.2:

The mystery of this staircase is that the information contained within it is—in effect—the full storyof where the primes are placed. This story seems to elude any simple description. Can we “tinkerwith” this staircase without destroying this valuable information?

12 Tinkering with the carpentry of the staircase of primes.

For starters, notice that all the risers of this staircase have unit length. That is, they contain nonumerical information except for their placement on the x-axis. So, we could distort our staircaseby changing (in any way we please) the height of each riser; and as long as we haven’t brought newrisers into—or old risers out of—-existence, and have not modified their position over the x-axis,we have retained all the information of our original staircase.

20

P(N)

250 500 750 1000

25

50

75

100

125

150

175

Figure 11.3:

P(N)

2500 5000 7500 10000

250

500

750

1000

1250

Figure 11.4:

P(N)

25000 50000 75000 100000

2500

5000

7500

10000

Figure 11.5:

21

A more drastic-sounding thing we could do is to judiciously add new steps to our staircase. Atpresent, we have a step at each prime number p, and no step anywhere else. Suppose we built astaircase with a new step not only at x = p for p each prime number but x = 1 and x = pn wherepn runs through all powers of prime numbers as well. Such a staircase would have, indeed, manymore steps than our original staircase had, but, nevertheless, would retain much of the quality ofthe old staircase: namely it contains within it the full story of the placement of primes and theirpowers.

A final thing we can do is to perform a distortion of the x-axis (elongating or shortening it, as wewish) in any specific way, as long as we can perform the inverse process, and “undistort” it if wewish. Clearly such an operation may have mangled, but hasn’t destroyed information irretrievably.

We shall perform all three of these kinds of operations eventually, and will see some great surprisesas a result. But for now, we will perform distortions only of the first two types. We are about tobuild a new staircase that retains the precious information we need, but is constructed accordingto the following architectural plan.

• We first build a staircase that has a new step precisely at x = 1, and x = pn for every primepower pn with n ≥ 1. That is there will be a new step at x = 1, 2, 3, 4, 5, 8, 9, 11, . . .

• Our staircase starts on the ground at x = 0 and height of the riser of the step at x = 1 willbe log(2π). The length of the riser of the step at x = pn the prime power pn will not be 1 (aswas the length of all risers in the old staircase of primes) but rather: the step at x = pn willhave the height of its riser equal to log p. So for the first few steps listed in the previous item,the risers will be of length N = log(2π), log 2, log 3, log 2, log 5, log 2, log 3, log 11, . . . Thesevertical dimensions might lead to a steeper ascent but no great loss of information.

Although we are not quite done with our architectural work, here is what our staircase lookslike, so far:

22

10 20 30 40 50

10

20

30

40

50

250 500 750 1000

250

500

750

1000

Figure 12.1: Illustration of the newly constructed staircase that counts weighted prime powers.

Notice that this new staircase looks, from afar, as if it were nicely approximated by the 45 degreestraight line, i.e., by the simple function x. In fact, we have—by this new architecture—yet a thirdequivalent way of formulating Riemann’s hypothesis:

The Riemann Hypothesis (third formulation)

This new staircase is essentially square root close to the 45 degree straight line, i.e., tothe simple function x.

13 To our readers of Part I.

We hope that you now understand our motivation for distorting our staircase, as we have justdone: the statement of the Riemann Hypothesis—admittedly as elusive as before—has, at least,been expressed elegantly and simply, given our new staircase that approximates (conjecturally withessential square root accuracy) a 45 degree straight line.

23

We have offered, so far, three equivalent formulations of the Riemann Hypothesis, all having todo with the manner in which the prime numbers are situated among all whole numbers. In doingthis, we hope that we have convinced you that—in the words of Don Zagier—primes seem to obeyno other law than that of chance and yet exhibit stunning regularity. This is the end of part I ofour booklet, and is largely the end of our main mission, to explain—in elementary terms—what isRiemann’s Hypothesis?

For readers who have studied Differential Calculus and who are happy with complex numbers, weshall go further and show that the pursuit of Riemann’s hypothesis may provide a key to deepstructure of primes, and to the nature of the laws that primes do obey.

Part II

14 How Calculus manages to find the slopes of graphs that haveno slopes

Differential Calculus, initially the creation of Newton and/or Leibniz, acquaints us with slopes ofgraphs,

a function

derivative

tangent line

-2 -1 1 2 3 4 5

-2

-1

1

2

3

4

5

Figure 14.1: A picture of a graph of a function (blue), a slope at a point (green), and the graph ofthe derivative (red).

24

Calculus explains to us how to calculate those slopes, and finally, shows us the power that we thenhave to answer problems we could not answer if we couldn’t compute those slopes.

Usually, in elementary Calculus classes we are called upon to compute slopes only of smoothgraphs—i.e., graphs that actually have slopes at each of their points, such as in the illustration justabove. What could Calculus possibly do if confronted with a graph that has jumps? For example:

0.5 1 1.5 2

0.5

1

1.5

2

Figure 14.2: A picture of a graph that is 1 up to a point and then 2 after that point.

The most comfortable way to deal with the graph of such a function is to just approximate it by anice smooth function

0.5 1 1.5 2

0.5

1

1.5

2

Figure 14.3: A picture of a smooth graph approximating the graph that is 1 up to some point xand then 1 after that point, the smooth graph being flat mostly.

and then take the derivative of that smooth function. Of course, this is just an approximation,so we might try to make a better approximation, which we do in each successive graph in the thefigure.

Notice, what is happening: as the approximation gets better and better, the derivative will be zeromostly, with a blip at the point of discontinuity, and the blip will get higher and higher. Now, earlymathematicians (Newton, Leibniz)—in replacing approximate speeds by instantaneous velocities

25

a = 0.20

0.5 1 1.5 2

1

2

3

4

5

6

a = 0.10

0.5 1 1.5 2

2.5

5

7.5

10

a = 0.05

0.5 1 1.5 2

5

10

15

20

a = 0.030.5 1 1.5 2

10

20

30

40

Figure 14.4: A picture of the derivative of a smooth graph approximating the graph that is 1 upto some point and then 0 after that point. In each case, the blue graph is 1 until 1− ε and 2 after1 + ε.

by passing to limits—had to wait a while before later mathematicians (e.g., Weierstrass) gave arigorous foundation for what they were doing. Happily however, this process we are discussing,of approximating discontinuous functions more and more exactly by smooth functions, and takingtheir derivatives to get the blip-functions as we have just seen in the graphs above led mathemati-cians (notably the French mathematician, Laurent Schwartz) to provide a mathematically rigorousfoundation (that we needn’t go into here) called the theory of distributions. Thanks to this theoryyou can, so to speak, go all the way to form what you can legitimately call the derivative of thefunction that is 0 before the point x and 1 afterwards; if we were to think of this as a function wewould be in trouble: it would have ∞ as a value at x, and—worse—it would be no use to us. Butthe modern remedy is to formulate a conceptual context in which it has a genuine existence, andcan be genuinely used in exactly the same way as the derivatives of smooth functions are used.The only change here is that the derivative of this discontinuous function has a status, not as afunction, but rather as a distribution (For people who are interested in pursuing this, see [***].Note: These “blip-functions” as we have been referring to them are more officially called Diracδ-functions, the adjective “Dirac” being in honor of the physicist who first worked effectively withthis concept, the “δ being the symbol he assigned to these objects, and the noun “function” shouldproperly be in quotation-marks for—properly speaking the Dirac δ-function is not—as we haveexplained above—a bona fide function but rather a distribution.

The curious aspect of the Dirac δ-function that has its blip at the point x is that it is flatly zero inany interval in the t-line that doesn’t include the point t = x. We say that this Dirac δ-function asits support at the point x.

So what happens when we take the derivative—in the sense of distributions—of a complicatedstaircase? For example:

Well, we would have blip-functions (alias: Dirac δ-functions) at each of points of discontinuity ofΨ(x); that is, at x = any power of a prime.

26

10 20 30 40 50

10

20

30

40

50

250 500 750 1000

250

500

750

1000

Figure 14.5: Illustration of the staircase Ψ(x) constructed in Part I that counts weighted primepowers.

27

10 20 30 40 50

10

20

30

40

50

Figure 14.6: Continuous approximation to the staircase Ψ(x) (in red) along with a plot (in blue)of the derivative of this approximation.

28

As we have hinted above, we lose no information if we further modify our staircase by distortingthe x-axis, replacing x by et (as long as we remember what it is we have done!) so—for reasonsthat we will only see after the next section—we reconfigure our distribution Ψ′(x) replacing it withthe distribution

Φ(t) := Ψ′(et)/et2 .

1 2 3 4

0.5

1

1.5

2

2.5

Figure 14.7: An approximation to the distribution Φ(t)

This distribution Φ(t) has a “blip” at t = 0 and also at each positive multiple of log p as p runsthrough the primes. It has its support at the discrete set of points t = 0 and t = multiples of log pas p runs through the primes.

Distributions with discrete support we will refer to as spike distributions because graphs offunctions that approximate them look like they have spikes around the points in their support.

The spike distribution Φ(t) will be the main actor, taking center stage, for the rest of Part II.

To be sure, none of the crucial valuable information of placement of primes is beyond retrieve, forthe blips constituting of this distribution (its support) would allow us to reconstruct the positionof the prime numbers among all numbers.

But we must explain our motivation for subjecting our poor initial staircase to this series of brutalacts of distortion to end up with the distribution Φ(t). For this we embark on a discussion ofspectral analysis.

29

15 What do primes, heat, and sound have in common?

[[todo (william)]] Aside. Here we want a really good, really concrete, neverthelessconcise, discussion of Fourier transforms, and spectra, with illustrations from theone of the web-pages that does a Fourier analysis of the sounds made by musicalinstrument, or those that analyze—say—vowel sounds in English. We might talk a bitabout white noise and pure tones. But all this is kept within very strict page limits,and—as always—written in a language completely comprehensible to anyone.

It may pay to make a digression to discuss the 1822 contribution of J.-B. Fourier, in a treatise(Theorie analytique de la Chaleur that, apparently, Lord Kelvin referred to as a “mathematicalpoem.” Fourier’s viewpoint is now used throughout the scientific world.

When applied to acoustics Fourier’s theory will aid in analyzing and/or synthesizing any sound:it will analyze the sound wave into its component pure frequencies. Also–if you are so disposed—you can synthesize that sound wave by recomposing it, given the information of its constituentfrequencies, and their amplitudes. If, say, the sound emitted as a “square wave” in the figurebelow,

0.5 1 1.5 2

-1.5

-1

-0.5

0.5

1

1.5

Figure 15.1: The square-wave with its various Fourier approximations

the first picture in the figure gives the so-called fundamental of the square wave; that is, thepure tone (a sine wave) whose frequency is the fundamental frequency of the square wave. Thesubsequent pictures show how you can achieve better and better approximations to that square

30

wave sound by adding more and more of its constituent higher frequency pure tones, these highercomponents being referred to as the harmonics of the square wave sound.

Fourier’s idea has settled in as the very language of optics, and electromagnetic phenomena: anysource of light is understood (analyzed) as comprised of its spectrum, i.e., the sum of purefrequencies–with their corresponding phases and amplitudes–that make up that light source. Sucha spectral analysis of, say, the light emitted from the Hydrogen atom, is both

• the signature of the hydrogen atom, in the sense that it is so characteristic of hydrogen that formthe spectrum alone you can tell if it is, or isn’t hydrogen that you are dealing with, and

• a crucial key to the deeper internal structure of hydrogen: the spectral lines are distanced, onefrom another, in proportion to the differences of squares of reciprocal whole numbers: a criticalclue, historically leading to the Bohr atom and beyond . . . .

The beauty of Fourier’s modus operandi is that it all arises from a single model wave (a sinewave). From this “model wave” both the fundamental and all the higher frequency harmonicsof any spectrum are fashioned; the harmonics are all simply sine waves as well, but modifiedappropriately in their frequency, phase, and amplitude. To synthesize any sound wave, say, youneed only know this numerical data (frequency, phase, and amplitude) of the fundamental, and ofeach of the higher harmonics.

16 Spike distributions that have discrete spectrum

Aside. Here we talk about distributions with discrete support; a ’spike distribution’is then a sum of δ-distributions supported on a discrete set of points. We talk aboutthe curious situation where the Fourier transform of one of these is again one, with afew clean examples. We note that any spike distribution that has the property thatits Fourier transform is again a spike also has the curious “error-correcting virtue”that if you ever forget a finite number of points of its support you can retrieve them.We need really interesting SAGE worksheets here.

17 The Spike distribution Φ(t); the Riemann Hypothesis (fourthformulation)

Here we should show a series of better and better approximations to the Fourier transform of Φ(t),noting that there are spikes at (or around, at least) the real numbers 14.13... etc.

We then blurt out:

31

There is an infinite sequence of real numbers θ1, θ2, θ3, . . . such that thedistribution Φ(t) is equal to the distribution defined as the summation

1 +∑∞

i= cos(θit)...

Aside. Here we will motivate the above equivalent statement of RH (that φ(t) is a sumof the cos(θt)’s for a discrete set Θ of θ’s. Discuss this amazing feature. Discuss thatif Θ exists at all, i.e., if RH is true, Θ has the “error-correcting virtue.” We begin tocompute it in a “worksheet” given knowledge of the primes, and we wonder whetherwe can find any other (nonbanal, of course) discrete set Θ′ that has the property thatthe sum of the cos(θ′t)’s for θ′ ∈ Θ′ is discrete.

Figure 17.1: Illustration of −∑1000

i=1 cos(sθi), where θ1 ∼ 14.13, . . . are the first 1000 frequencies.The red dots are at the prime powers pn, whose size is proportional to log(p).

32

Figure 17.2: Illustration of −∑1000

i=1 cos(sθi) in the neighborhood of a twin prime. Notice how thetwo primes 29 and 31 are separated out by the Fourier series, and how the prime power 25 appears.

33

Figure 17.3: Fourier series illustration from 1000 to 1030

18 The undistorted staircase: returning to P (N)

We have been dealing in this Part II of our book with Φ(t) a distribution that—we said—containsall the essential information about the placement of primes among numbers. We have given aclean restatement of Riemann’s hypothesis, the fourth restatement so far, in term of this Φ(t). ButΦ(t) was the effect of a series of recalibrations and reconfigurings of the original untampered-withstaircase of primes. A test of whether we have strayed from our original problem—to understandthis staircase—would be whether we can return to the original staircase, and “reconstruct it” soto speak, soley from the information of Φ(t)—or equivalently, assuming the Riemann hypothesisas formulated in the previous section—can we construct the staircase of primes P (N) solely fromknowledge of the sequence of real numbers θ1, θ2, θ3, . . .

The answer to this is yes, and is given very beautifully by Bernhard Riemann himself in his famous1859 article cited above.

Think of Riemann’s smooth curve R(X) as the fundamental approximation to P (N). Riemannoffered much more than that in his wonderful 1859 article.

WILLIAM: More facsimile photos of Riemann ’s manuscript

Bernhard Riemann provided an exact formula for P (N) (a formula reminiscent of Fourier’s analysisof functions as constituted out of sines) that analyzes and/or synthesizes the staircase of primes.

34

Figure 17.4: The Fourier transform of the distribution supported on prime powers, i.e., which isδ log(p)

pn/2 at the prime power pn.

35

0.5 1 1.5 2 2.5 3

-0.08

-0.06

-0.04

-0.02

0.02

0.04

0.06

0.08

Figure 17.5: Illustration ofθ sin(tθ) + 1

2 cos(tθ)θ2 + 1/4

, for θ ∼ 14.13, which is the wave component of the

first Fourier approximation to Φ(t)

36

Figure 17.6: The first two wave components that appear in the Fourier series of Φ(t), areθ sin(tθ)+ 1

2cos(tθ)

θ2+1/4for θ ∼ 14.13, 21.02

0.5 1 1.5 2 2.5 3

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 17.7: The first term of the Fourier series of Φ(t) iset/2θ sin(tθ) + 1

2et/2 cos(tθ)

θ2 + 1/4, where

θ ∼ 14.134725

37

Figure 17.8: The first two components of the Fourier series of Φ(t) (including the exponential

factors) are et/2 · θ sin(tθ)+ 12

cos(tθ)

θ2+1/4for θ ∼ 14.13, 21.02

Figure 17.9: Fourier approximations to the step function Ψ obtained using 1, 2, 5, and 30 frequen-cies.

38

Figure 17.10: Fourier approximation to the step function Ψ obtained using 30 frequences andgraphed from 4 to 5.

Figure 17.11: Fourier approximation to the step function Ψ obtained using 30 frequences.

39

Riemann started with a specific smooth function, that we will refer to as R(X), a function theRiemann offered, just as Gauss offered his Li(X), as a candidate smooth function approximatingthe staircase of primes3

Riemann’s R(X) seems to be a better approximation to P (N) than Gauss’s Li(X) :

WILLIAM: maybe a comparison of the three graphs π(X), Li(X) and R(X) showingR(X) closer?

Just as Gauss’s Li(X) has—in its favor— a probabilistic plausibility argument that might suggestthat it is a good approximation to the number of primes, Riemann’s R(X) has, in fact, a morerefined probabilistic plausibility argument.

NOTE: maybe a comparison of the three graphs π(X), Li(X) and R(X) showing R(X)closer?

Not only did Riemann provide a “fundamental” (that is, a smooth curve that is an astoundinglyclose to P (N)) but he viewed this as just a starting point, for he gave the recipe for providing aninfinite sequence of corrective terms—call them Riemann’s harmonics; we will denote the first ofthese “harmonics” C1(X), the second C2(X), etc. Riemann gets his first corrected curve, R1(N),from R(N) by adding this first harmonic to the fundamental,

R1(N) = R(N) + C1(N),

he gets the second by correcting R1(N) by adding the second harmonic

R2(N) = R1(N) + C2(N),

and so onR3(N) = R2(N) + C3(N),

and in the limit provides us with an exact fit.

The Riemann Hypothesis, if true, would tell us that these correction terms C1(X), C2(X), C3(X), . . .are all square-root small, and all the successively corrected smooth curves

R(X), R1(X), R2(X), R3(X), . . .

are good approximations to P (N).3Gauss’s guess, usually denoted Li(x), is Li(x) =

R∞2

dx/log(x), while Riemann’s guess is

R(x) =

∞Xn=1

µ(n)

nLi(x

1n ),

where µ(n) is the Moebius function.

40

The elegance of Riemann’s treatment of this problem is that the corrective terms Ci(N) are allmodelled on the fundamental R(N) and are completely described if you know the sequence of realnumbers θ1, θ2, θ3, . . . of the last section.

PUT FIGURES N,O,P, Q HERE;–this should be Williams graphs of Rk’s, for k =1, 29, 50, 100 to show the emergence of a kind of stepwise shape of the smooth curves inquestion.

(2) He provided an extraordinary recipe that allows us to work out the harmonics,

C1(X), C2(X), C3(X), . . .

without our having to consult, or compute with, the actual staircase of primes. As with Fourier’smodus operandi where both fundamental and all harmonics are modeled on the sine wave, butappropriately calibrated, Riemann fashioned his higher harmonics, modeling them all on a singlefunction, namely his initial guess R(X).

Aside. We use Riemann’s function R(X), and discuss how to reconstruct the originalstaircase of primes by the Rk’s. [[todo: put riemann Rk worksheets 6,8,36,37 here.]]

Figure 18.1: The function R10 approximating the staircase of primes up to 500

[*** All this needs to be written***]

41

Figure 18.2: The function R29 approximating the staircase of primes up to 100

42

19 William: some alternative text about Riemann

From Riemann’s smooth curve to the staircase of primes. Approximations are all well andgood, but exact formulas are even better. Can we get a better formula than Riemann’s R(N) byadding to it a clever, simple to calculate, correction term? And slightly more ambitiously,

Can we achieve an exact formula for the precise accumulation of primes, P (N),–a formula ratherthan a mere approximation–by making a clean, infinite, sequence of explicit corrections to Riemann’scurve?

The answer to this is given very beautifully by Bernhard Riemann himself in his famous 1859 articlecited above.

Think of Riemann’s smooth curve R(X) as the fundamental approximation to P (N). Riemannoffered much more than that in his wonderful 1859 article.

(1) He provided an exact formula for P (N) (reminiscent of Fourier) that analyzes and/or synthe-sizes the staircase of primes, starting with his “fundamental,” R(X) and correcting it successivelyby stipulating an infinite sequence of correction terms– call these correction terms Riemann’s har-monics; we will denote the first of these “harmonics” C1(X), the second C2(X), etc. Riemann getshis first corrected curve, R1(N), from R(N) by adding this first harmonic to the fundamental,

R1(N) = R(N) + C1(N),

he gets the second by correcting R2(N) by adding the second harmonic

R2(N) = R1(N) + C2(N),

to the first corrected and so on. He provided an extraordinary recipe that allows us to work outthese harmonics,

C1(X), C2(X), C3(X), . . .

without our having to consult, or compute with, the actual staircase of primes. As with Fourier’smodus operandi where both fundamental and all harmonics are modeled on the sine wave, butappropriately calibrated, Riemann fashioned his higher harmonics, modeling them all on a singlefunction, namely his initial guess R(X).

The Riemann Hypothesis, if true, would tell us that these correction terms C1(X), C2(X), C3(X), . . .are all square-root small, and all the successively corrected smooth curves

R(X), R1(X), R2(X), R3(X), . . .

are good approximations to P (N), and in the limit provide us with an exact fit.

For people who know complex numbers, for k = 1, 2, . . . the k−th corrective term is given by

Ck(X) = −R(√XXiθk)

43

where these mysterious exponents θ1, θ2, θ3, . . . are the same real numbers discussed in the previoussection: the spectrum of the distribution Φ(t).

PUT Williams graphs of Rk’s, for various K (k = 1, 29, 50, 100?) to show the emergenceof a kind of stepwise shape of the smooth curves in question.

Best of all, Riemann provided an extraordinary recipe that allows us to work out the sequence ofreal numbers harmonics, θ1, θ2, θ3, . . . and therefore the corrective terms

C1(X), C2(X), C3(X), . . .

Part III

20 Further alternative text here

Aside. Here we point out that even though the mere existence of the θ’s implied thesquare root approximation info that we began our discussion with, we computed the θ’sonly assuming RH and knowing the placement of primes. So if we are ever to use the θ’sto tell us more precise things about the placement of primes, we need an independentway of computing them. Also, if we had an independent way of computing them, wesimply have We then move towards discussion of the Riemann zeta function.

[[Insert color pics here.]]

In the previous sections we gave a description of the spectrum of primes that provided the keyto calculating the tinkered-with staircase of primes. Suppose you didn’t want to tinker with thecarpentry of that staircase. . . . Half a century after Gauss’s initial exploration, Bernhard Riemannproduced a slight variant of Gauss’s approximating curve,1 which has some heuristic reason to bean ever-so-small improvement in the approximating accuracy [one can explain Riemann’s curve alsoby consideration of probability]. Riemann’s curve, which we will denote R(X), seems to predict theactual number of primes to an astounding accuracy. For example, there are precisely 50, 847, 534prime numbers under a billion and Riemann’s curve would predict a number that undercounts this50, 847, 534 by a tiny 79. If you want to compare Gauss’s curve, Riemann’s curve, and the actualstaircase of primes, for the range of numbers less than or equal to 50, 000, there is no problem:Figure 4, given the thickness of the printed curve there, is a picture of any of these three curves,they are so close! But despite Figure 4, appearances can be deceiving, for –for all we know– bothcurves, Gauss’s and Riemann’s, may eventually stray far from the precious staircase of primes thatthey are supposed to be approximating. Riemann’s curve R(X) gives us an estimate for P (N), thenumber of primes less than or equal to N .

For a number N , how close, really, is Riemann’s estimate, R(N), to P (N), the true number ofprimes?

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With this new piece of vocabulary, “ approximation with square-root error, ” alias “good approxi-mation,” we can frame a clean question that might make more precise what we have seen visuallyas the miraculous convergence, in Figure 4, of the graphs of Riemann’s curve and the true staircaseof primes:

(*) Is R(N) a good approximation to the actual accumulation of primes, P (N)?

We have shaped this question inspired by a certain amount of things, which we haven’t gone into inthis lecture, but which are already mathematically proven; namely, R(N) is known to be no betteran approximation to the accumulation of primes, than an approximation with square-root error.On the other hand, R(N) is also known not to be wildly off the mark: the ratio of R(N) to P (N)tends to 1; the essential mathematical content of this– but not the way we have just expressedit– was already guessed by Gauss when he was only fifteen, and finally proved by Hadamard andindependently de la Vallee-Poussin, close to a century later, and so is a bona fide theorem, notmerely a hypothesis!

Part III

21 The Riemann Zeta-Function; and Riemann’s Hypothesis (fifthversion)

Here we develop a tiny bit of the traditional route of exposition to RH. Namely, interms of the successive nontrivial zeroes of Riemann’s zeta-function. There are infinitely manyof these nontrivial4 zeroes of the ζ-function. If you want to see tables of the first 100, 000 of thesezeroes lovingly calculated to an accuracy within 3 · 10−9, consult Andrew Odlyzko’s tables:

www.dtc.umn.edu/ odlyzko/zeta tables/. The first three of these zeroes are:

ρ1 =12

+ 14.134725 . . . i ρ2 =12

+ 21.022040 . . . i ρ3 =12

+ 25.01 . . . i etc..]

I hope that it is OK if I don’t explain this at all, for it does require calculus but we can think ofthese “zeroes” as being a certain precious infinite sequence of quantities, that contain within themthe secret of the exact determination of the pattern of prime numbers. (You can squint at the wayRiemann wrote out the zeta-function by trying to make out the formula on the first page of hismanuscript, in figure 5a.)

You will notice the curious persistence of the 12 ’s in the exponents of the three zeroes listed above.

Riemann called attention to this phenomenon, which we now know holds for all 1029.9 billion zeroes

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that have been tabulated as of Feb. 18, 2005! The 12 in the exponent of a correction term is what

guarantees that the correction term is indeed “square- root small.” We are ready, then, for anotherequivalent formulation of the Riemann Hypothesis, this being–in fact–the more traditional way ofexpressing it:

The Riemann Hypothesis (fifth formulation)

The Riemann Hypothesis is equivalent to the statement that all the nontrivial zeroes ofthe Riemann zeta-function lie on the line 1

2 + iy in the complex plane

That a simple geometric property of these zeroes (lying on a line!) is directly equivalent to suchprofound (and more difficult to express) regularities among prime numbers suggests that these ze-roes and the parade of Riemann’s corrections governed by them–when we truly comprehend theirmessage–may have lots more to teach us, may eventually allow us a more powerful understanding ofarithmetic. This infinite collection of complex numbers, i.e., the nontrivial zeroes of the Riemannzeta function, plays a role with respect to P (N) rather like the role the spectrum of the Hydrogenatom, plays in Fourier’s theory. Are the primes themselves no more than an epiphenomenon, be-hind which there lies, still veiled from us–a yet-to-be-discovered, yet-to-be-hypothesized, profoundconceptual key to their perplexing orneriness. Are the many innocently posed, yet unanswered,phenomenological questions about numbers–such as in the ones listed earlier– waiting for our dis-covery of this deeper level of arithmetic? Or for layers deeper still? Are we, in fact, just at thebeginning?

These are not completely idle thoughts, for a tantalizing analogy relates the number theory wehave been discussing to an already established branch of mathematics–due, largely, to the workof Alexander Grothendieck, and Pierre Deligne–where the corresponding analogue of Riemann’shypothesis has indeed been proved. . .

22 Some fragments to be either used or discarded

(2) If f(x) and g(x) are real-valued functions of a real variable x such that for any ε > 0 bothof them take their values between x1−ε and x1+ε for x sufficiently large, then say that f(x) andg(x) are good approximations of one another if, for any positive ε the absolute values of theirdifference is less than x

12+ε for x sufficiently large. The functions Li(x) and R(x) of end-note (1)

are good approximations of one another.

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(3) People who know that these correction terms are index by the nontrivial zeroes of the Riemannzeta-function may well ask how I propose to order them if RH is false; the following prescriptionwill do: order them in terms of (the absolute value of) their imaginary part, and in the unlikelysituation that there is more than one zero with the same imaginary part, order zeroes of the sameimaginary part by their real parts, going from right to left.

23 Glossary

Aside. Here we will be giving all the connections with the standard literature andcoventional terminology that we restrained ourselves from giving in the text itself.Our aim is not even to mention complex numbers in the text, so no Riemann zetafunction either, but in this glossary we can talk about ζ(s) its zeroes, and these beingthe frequencies mentioned in the text. For the moment the list of entries is the followingbut it will expand.

P (N) = π0(N), Q(N) = ψ0(N), log, exp, δ, distributions, RSA cryptography, Mersenneprime, Li(x), random walk, spectrum, harmonic, fundamental, frequency, phase, am-plitude, band-pass, complex numbers, complex plane, Riemann Zeta function, zeroesof zeta.

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