The statement of the Riemann Hypothesis – admittedly as elusive as before – has, at least, been expressed elegantly and more simply, given our new staircase that approximates (conjecturally with essential square root accuracy) a 45 degree straight line.

We have offered two equivalent formulations of the Riemann Hypothesis, both having to do with the manner in which the prime numbers are situated among all whole numbers.

In doing this, we hope that we have convinced you that – in the words of Don Zagier – primes seem to obey no other law than that of chance and yet exhibit stunning regularity. This is the end of Part I of our book, and is largely the end of our main mission, to explain – in elementary terms – what is Riemann's Hypothesis?

For readers who have at some point studied differential calculus, in Part II we shall discuss Fourier analysis, a fundamental tool that will be used in Part III where we show how Riemann's hypothesis provides a key to some deeper structure of the prime numbers, and to the nature of the laws that they obey. We will – if not explain – at least hint at how the above series of questions have been answered so far, and how the Riemann Hypothesis offers a surprise for the last question in this series.

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 approximations 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 synonymously, 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 amere 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 in 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 toq(X) if for any given exponent greater than 0.5 (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 Xraised to that exponent (e.g. X0.501, < X0.5001, etc.).

For starters, notice that all the (vertical) risers of this staircase (Figure 18.1 above) have unit height. 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.

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

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

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

1. • We first build a staircase that has a new step precisely at x = 1, and x = pn for every prime power pn with n≥ 1. That is, there will be a new step at x= 1, 2, 3, 4, 5, 7, 8, 9, 11, . . .

Mathematicians seems to agree that, loosely speaking, there are two types of mathematics: pure and applied. Usually – when we judge whether a piece of mathematics is pure or applied – this distinction turns on whether or not the math has application to the “outside world,” i.e., that world where bridges are built, where economic models are fashioned, where computers churn away on the Internet (for only then do we unabashedly call it applied math), or whether the piece of mathematics will find an important place within the context of mathematical theory (and then we label it pure). Of course, there is a great overlap (as we will see later, Fourier analysis plays a major role both in data compression and in pure mathematics).

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., the question raised in this book, to find a smooth curve that is a reasonable approximation to the staircase of primes). And only as things develop is it discovered that there are payoffs in many unexpected directions, some of these payoffs being genuinely applied (i.e., to the practical world), some of these payoffs being pure (allowing us to strike behind the mask of the mere appearance of the mathematical situation, and get at the hidden fundamentals that actually govern the phenomena), and some of these payoffs defying such simple classification, insofar as they provide powerful techniques in other branches of mathematics. The Riemann Hypothesis – even in its current unsolved state – has already shown itself to have all three types of payoff.

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 given by a simple analytic formula? 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?