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What is Riemann’s Hypothesis?

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—with or 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, that have the Riemann Hypothesis as their main topic. A reader of these books will get a fairly rich picture of the personalities engaged in the pursuit, and of related mathematical and historical issues.

“There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly 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 the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the 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 Number are “the principles of all things,” the elements of number being more basic than the Empedoclean physical elements earth, air, fire, water. To think about number is to get close to the architecture of “ what is.”

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

Ren´e Descartes, almost four centuries ago, expressed the hope that there soon would be “almost nothing more to discover in geometry.” Contemporary physicists dream of a “final theory.” But despite its venerability and its great power and beauty, the pure mathematics of numbers may still be in the infancy of its development, with depths to be explored as endless as the human soul, and never a final theory.

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

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 spelling out her name. The “bachelor” found

“a great difficulty in their composition because the number of letters in her name was 17, and if he made four Castilian stanzas of four octosyllabic lines each, there would be one letter too many, and if he made the stanzas of five octosyllabic lines each, the ones called d´ecimas 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 of the number 17 to division.

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

Prime numbers, despite their primary position in our modern understanding of number, were not specifically doted over in the ancient literature before Euclid, at least not in the literature that has 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, which is surprising to me given the intense interest Plato had in mathematical developments; and they make an occasional appearance in the writings of Aristotle, which is not surprising, given Aristotle’s emphasis on the distinction between the composite and the incomposite. “The incomposite is prior to 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 now understood to be.

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

3 What are prime numbers?

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

having over 150 decimal digits are used to keep our bank transactions private. This ubiquitous use to which these giant primes are put depends upon a very simple principle: it is much easier to multiply numbers together than to factor them. If you had to factor, say, the number 143 you might scratch your head for a few minutes before discovering that 143 is 11 × 13. But if you had to multiply 11 by 13 you would do it straightaway. Offer two primes, say, P and Q each with more than 100 digits, to your computing machine and ask it to multiply them together: you will get their product N = P × Q with its 200 or so digits in nanoseconds. But present that number N to any 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 guaranteed^1 failure!

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

p = 2^32582657 − 1

is a prime number, this number having 9, 808 ,358 digits! There are, in fact, infinitely many prime numbers, as was proved in Euclid’s Elements as follows. Suppose there are only finitely many primes p 1 ,... , pn. Let n = p 1 p 2 · · · pn + 1. Then n is divisible by some prime, but no pi divides n, which is contrary to our assumption that p 1 ,... , pn is the complete list of primes.

The number p displayed above is the largest prime we know, where by “know” we mean that we know it so explicitly that we can compute things about it. For example, the last two digits of p are

71. Of course p is not the largest prime number since there are infinitely many primes, e.g., the next prime q after p is a prime. But there is no obvious way to compute anything about q. For example, what is the last digit of q in its decimal expansion?

Eratosthenes, the mathematician from Cyrene (and later, librarian at Alexandria) explained how to 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 back to the beginning of our sequence of numbers and circle the first number that is neither circled nor crossed out (that would be, of course, the 3); then cross out all the other multiples of 3. This gives the pattern: go back again to the beginning of our sequence of numbers and circle the first number that is neither circled nor crossed out; then cross out all of its other multiples. Repeat this pattern until all the numbers in our sequence are either circled, or crossed out, the circled ones being the primes.

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 follow Eratosthenes’ 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 ,...

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

4 Questions about primes that any person might ask

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

For example, are there infinitely many pairs of primes whose difference is 2? The sequence on the page 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 than 1 , 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 every even number greater than 2 a sum of two primes? Answer: unknown. Are there infinitely many primes 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 method that answers that question without, in some form or other, running through each of the successive numbers 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, of course, that the even numbers and the odd numbers are nicely and simply distributed: after every odd number comes an even number, after every even, an odd, there is an equal number of odd number as even numbers less than any given odd number; and there may be nothing else of interest to say about the matter. Things change considerably, though, if we focus our concentration on multiplicatively even numbers and multiplicatively odd numbers.

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

There are 455, 052 , 512 primes less than ten billion; i.e., 10,000,000,000 (so we might say that the chances 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 look at a few graphs, to get a sense of why that might be so. There are a hundred numbers less than or equal to 100, a thousand numbers less than or equal to 1000, etc.: the shaded bar graph that looks like a regular staircase, each step the same length as each riser, climbing up at, so to speak, a 45 degree angle, counts all numbers up to and including N.

Following Eratosthenes, we have sifted those numbers, to pan for primes. Our first move was to throw out roughly half the numbers (the even ones!) after the number 2. The cross-hatched bar graph in this figure which is, with one hiccup, a regular staircase climbing at a smaller angle, each step twice the lengths of each riser, illustrates the numbers that are left after one pass through Eratosthenes’ sieve, which includes, of course, all the primes. So, the chances that a number bigger than 2 is prime is at most 1 in 2. Our second move was to throw out a good bunch of numbers bigger than 3. So, the chances that a number bigger than 3 is prime is going to be even less. And so it goes: with each Eratosthenian move in our sieving process we are winnowing the field more extensively, 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 staircase of primes. Its height above any number N on the horizontal line records the number of primes less than or equal to N , the accumulation of primes up to N. Refer to this number as P (N ). So P (2) = 1, P (3) = 2, P (30) = 10; of course, if you believed some of the data above you could plot a 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 of all whole numbers after we have removed the even numbers greater than 2, and the multiples of 3 greater than 3 itself:

From this graph you can see that if you go “out a way” the likelihood that a number is a prime is less 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 line the 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 for N = 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 the eye. 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 just as faithful a reproduction of the staircase of primes as the typographer’s art can render, for there are about 8, 000 tiny steps and risers in this curve, all hidden by the thickness of the print of the drawn curve in the figure. It is already something of a miracle that we can approximately describe

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

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

[[TODO (william): Insert a discussion here about how believing RH is crucial to doing many algebraic number theory calculations. E.g., complexity analyses for algorithms like factoring depend on RH. Also, standard class group algorithms are vastly faster if we assume RH (or even more!), so number theorists do so all the time when doing numerical experiments (though of course they try to remove 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 understand the above!”]]

8 A probabilistic “first” guess

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

Gauss was an inveterate computer: he wrote in his 1849 letter that there are 216, 745 prime numbers less than three million (This is wrong: the actual number of these primes is 216816). Gauss’s curve predicted that there would be 216, 971 primes– a miss, Gauss thought, by 226 (but actually he was closer than he thought: the correct miss is a mere 161; not as close as the recent US elections, but pretty 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” continue for arbitrarily large numbers? and, the (evidently prior) question: what counts as a good fit?

Area ~ 29.

25 Primes

1/log(x) 25 50 75 100

-0.

-0.

0.8 Area ~ 176.

168 Primes

1/log(x) 250 500 750 1000

-0.

-0.

Figure 8.1: The expected tally of the number of primes < X is equal to the area underneath the graph 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 within a hundred, let us celebrate this kind of accuracy by saying that you have made an approximation with 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 you get it right to within a thousand, let’s agree that—again—you have made an approximation with square-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 you have 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 a single error term,

Error term = Exact Value - Our “good approximation”

but rather for infinite sequences of estimates of error terms. Generally, if you are interested a numerical 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), to this quantity let us say that qapprox(X) is essentially a square-root accurate approximation to q(X) if for any given exponent greater than 1/2 (you choose it: 0.501, 0.5001, 0.50001,... for example) and for large enough X— where the phrase “large enough” depends on your choice of exponent—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. < X^0.^501 , < X^0.^5001 , etc.).

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 both functions X/logX (as in our “first formulation”) and Li(X) (as in this “second formulation”) are respectably close approximations to π(X). We know, for example, that for any number greater than 1 that you give (we will illustrate this by imagining that you gave the number 1.000000001) if you take X sufficiently large (and how large “sufficiently large” is depends on the number that you gave) π(X) is no greater than 1.000000001 times Li(X), and Li(X) is no greater than 1. 000000001 times π(X). The same is true if Li(X) is replaced by X/ log X. These facts are known as The Prime Number Theorem, a very hard-won piece of mathematics! (See the Appendix for a discussion of this 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’s magnificent 8-page paper, “On the number of primes less than a given magnitude,” published in

Figure 10.1: From Riemann’s Manuscript

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

Riemann’s hypothesis, as it is currently interpreted, turns up as relevant, as a key, again and again in different parts of the subject: if you accept it as hypothesis you have an immensely powerful tool 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 these formulations being provably equivalent to any of the others.

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

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 by Marcus du Sautoi, for we feel that it captures the sense that there is a deeply hidden architecture to the graphs that compile the number of primes (up to N ) and also because—in a bit—we will be tinkering 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 story of where the primes are placed. This story seems to elude any simple description. Can we “tinker with” 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 no numerical information except for their placement on the x-axis. So, we could distort our staircase by changing (in any way we please) the height of each riser; and as long as we haven’t brought new risers 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.