The slope a Gauss sum leaves behind

Close in height and nowhere close in slope showed that uniform convergence passes continuity to a limit and passes nothing about slopes. Its last example was a series of the kind that makes the point most sharply: add sine waves whose frequencies grow faster than their amplitudes shrink, and the sum is continuous while every partial sum is smooth and the limit may be smooth nowhere. Karl Weierstrass made the first proved example in 1872, and in presenting it to the Berlin Academy he said that Bernhard Riemann had already proposed another, in lectures around 1861:

$$R(x) = \sum_{n=1}^{\infty} \frac{\sin(n^2 x)}{n^2} = \sin x + \frac{\sin 4x}{4} + \frac{\sin 9x}{9} + \cdots.$$

Riemann left no proof, and Weierstrass, unable to supply one, built a different function for which he could. The terms are bounded by $1/n^2$, so the series converges uniformly and $R$ is continuous. Its formal derivative, $\sum \cos(n^2 x)$, converges nowhere. Whether $R$ has a derivative at any point turned out to be a hard question, open for about a century, and the answer is that Riemann’s candidate fails, but only just: it has a derivative on a dense set of points, every one a rational multiple of $\pi$, and the derivative there is always exactly $-\tfrac12$.

A Fourier series with square frequencies #

Riemann’s function is a Fourier series, and an unusual one. An ordinary Fourier series, like the one that builds a square wave from round ones, uses every frequency $1, 2, 3, \ldots$ with coefficients chosen to fit a given shape. Riemann’s uses only the square frequencies $1, 4, 9, 16, \ldots$, with coefficients that fall like the reciprocal of the frequency. Weierstrass’s own function uses geometric frequencies $b, b^2, b^3, \ldots$, and the gaps between them are what make his proof work: each new wave is so much faster than all the earlier ones that it dominates the increment at its own scale. Square frequencies grow more slowly, the gaps between them are proportionally smaller, and the waves interfere with one another over a range of scales rather than taking turns. That interference is why Weierstrass could not prove Riemann’s function rough and why, at special points, it cancels into smoothness.

Rough functions are not rare. Baire’s category argument, which a limit can jump at every fraction introduced, shows that in the space of continuous functions the ones with a derivative at even a single point form a negligible set. What is rare is a function given by a formula whose differentiability can be decided point by point. Riemann’s is one of very few, and the decision runs entirely through arithmetic. The waves with square frequencies line up at $\pi p/q$ according to the residues of $n^2$ modulo $2q$, and those residues are what a Gauss sum adds.

Computing a function nobody can evaluate naively #

Every figure here rests on one computation: the increment $R(x_0 + h) - R(x_0)$ for small $h$, accurate enough to see how it scales. That is harder than it looks. Truncating the series at $N$ terms leaves a tail of size up to $1/N$, which swamps an increment of size $10^{-6}$ unless $N$ is in the millions. And at $N = 10^6$ the argument $n^2 x_0$ is about $3 \times 10^{12}$. A double-precision $x_0$ carries an error of about $10^{-16}$ in relative terms, so $n^2 x_0$ is wrong by about $3 \times 10^{-4}$ — and the sine of it is wrong by the same amount, a thousand times the signal.

Both problems have clean answers. For the increment, the tail beyond $N$ is a sum of terms with essentially random phases once $n^2 h$ is large, so its size is about $N^{-3/2}$ rather than $1/N$; a million terms leave an error of about $10^{-9}$. For the phases, the points that matter are rational multiples of $\pi$. At $x_0 = \pi p/q$ the angle $n^2 x_0$ modulo $2\pi$ depends only on $n^2 p$ modulo $2q$, which is a computation on small whole numbers with no rounding at all. For an irrational point such as $\pi(\sqrt5 - 1)/2$ the multiplier is stored to 128 binary places and multiplied exactly by $n^2$ before reduction. The double-precision sine then never sees an argument larger than $2\pi$ plus $n^2 h$, and $n^2 h$ is computed with full relative accuracy.

Three points under the microscope #

At most points a graph like this one has no visible structure, and the only way to see what happens at a point is to zoom. The zoom has to be rescaled, though. If the increment behaves like $|h|^\alpha$, a window of width $w$ shows heights of order $w^\alpha$, and to compare windows the vertical scale must be divided by that. Dividing by $\sqrt w$ keeps a square-root cusp at a fixed size.

The three points behave differently. At $2\pi/3$ the picture is a spike: the function drops away like $\sqrt{|h|}$ on both sides, and the spike has the same shape at both magnifications. That is a cusp, and there is no derivative. At $\pi/2$ the picture is lopsided — flat to the right, a square-root drop to the left — and the shape again survives magnification, so there is a one-sided cusp and still no derivative. At $\pi$ the curve gets flatter as the window shrinks. The increment there is shrinking faster than $\sqrt{|h|}$; under this rescaling a straight line of any slope eventually lies flat, and that is what the bottom-left panel shows.

So the zoom separates two kinds of point: those where something of order $\sqrt{|h|}$ is left, and those where it is not. The question is which points are which, and what the size of the square-root term is when it is there.

The square root has a coefficient, and it is a Gauss sum #

Take each fraction $p/q$ in lowest terms with $q$ at most 7 and $0 \le p/q < 2$ — thirty-six points spread over one period — and measure the coefficient of $\sqrt{|h|}$ on each side: $(R(x_0 + h) - R(x_0))/\sqrt h$ at $h = 10^{-6}$ and at $h = -10^{-6}$, after removing the term $-h/2$ that turns out to be present everywhere.

The numbers that come out have a pattern. At $0$ the coefficient is $1.2533$ on the right and $-1.2533$ on the left; that is $\sqrt{\pi/2}$, to four places. At $\pi/2$ it is $0$ on the right and $-1.2533$ on the left. At $2\pi/3$ it is $-0.7236$ on both sides, which is $\sqrt{\pi/2}$ divided by $\sqrt3$. At $2\pi/5$ it is $0.5605$ and $-0.5605$, which is $\sqrt{\pi/2}$ divided by $\sqrt5$. The square roots of the denominators are the signature of a quadratic Gauss sum,

$$g(p, q) = \frac{1}{2q}\sum_{k=0}^{2q-1} e^{i\pi p k^2/q},$$

the average of the points on the unit circle at angles $\pi p k^2/q$. Its size is $1/\sqrt q$ or zero, according to the parities of $p$ and $q$ — the same sums that square to a prime in the proof of quadratic reciprocity.

The rule the figure checks is

$$R\!\left(\tfrac{\pi p}{q} + h\right) - R\!\left(\tfrac{\pi p}{q}\right) \approx \sqrt{\tfrac{\pi |h|}{2}}\,\bigl(\operatorname{Re} g - \operatorname{Im} g\bigr) - \tfrac{h}{2}$$

for $h > 0$, and the same with $\sqrt{\pi |h|/2}\,(\operatorname{Re} g + \operatorname{Im} g)$ subtracted instead for $h < 0$. Every one of the seventy-two measured coefficients matches its prediction to within $0.01$. The Gauss sum is zero exactly when $p$ and $q$ are both odd, and at those points the square-root term vanishes on both sides. What is left is the linear term $-h/2$ — a derivative of $-\tfrac12$.

Why the slope is minus one-half #

The rule has a reason, and the reason explains the $-\tfrac12$ as well. The formal derivative of $R$ is $\sum_{n \ge 1} \cos(n^2 x)$, which is the real part of $\sum_{n \ge 1} e^{in^2 x}$. Add in the missing $n = 0$ term and the negative $n$, and that is half of Jacobi’s theta function, $\theta(x) = \sum_{n \in \mathbb{Z}} e^{i n^2 x}$, minus one-half. Theta is not a function in the ordinary sense on the real line, but it has a precise local structure. Near $x = \pi p/q$ the identity that relates theta at $x$ to theta at $-1/x$, the one that makes it a modular form, says it behaves like the Gauss sum $g(p,q)$ times a singularity of size $|h|^{-1/2}$, plus terms that are much smaller.

Integrating the derivative turns $|h|^{-1/2}$ into $|h|^{1/2}$ — the square-root cusp — with the Gauss sum as its coefficient. And the constant $-\tfrac12$ from the missing $n = 0$ term integrates to $-h/2$, the linear term that is present at every point. Where the Gauss sum vanishes, the singular part of theta vanishes with it, and the only thing left at leading order is that constant. The derivative of $-\tfrac12$ is the ghost of the term $n = 0$, which the series leaves out because $\sin(0)/0^2$ means nothing.

This is also why Riemann’s function is self-similar under magnification. Theta’s modular symmetry maps a neighbourhood of one rational point onto a neighbourhood of another, so the picture at $2\pi/3$ is the picture at $0$ transformed, shrunk by a factor set by the denominator. Hans Duistermaat made this precise in 1991. The $1/\sqrt q$ in the coefficients is the shrinking.

The slope, measured #

The rule predicts a derivative of $-\tfrac12$ at every $\pi \times \text{odd}/\text{odd}$. Measured directly, by symmetric difference quotients:

At large $h$ the quotients wander widely, because a window of width $0.1$ around $3\pi/5$ contains other fractions with small denominators and their cusps. As $h$ shrinks the window excludes them one by one. By $h = 10^{-4}$ all four quotients are within a few hundredths of $-\tfrac12$, and at the smallest $h$ they agree with it to two decimal places. $5\pi/7$, whose Gauss-sum structure involves the larger denominator 7, settles last.

Joseph Gerver proved in 1970 that $R$ has derivative exactly $-\tfrac12$ at every point $\pi(2a+1)/(2b+1)$, overturning the general belief that Riemann’s function was nowhere differentiable. G. H. Hardy had shown in 1916 that $R$ has no derivative at any irrational multiple of $\pi$ and at many rational ones, and believed the rest would follow. Gerver completed the picture in 1971 by showing that every other point fails.

How rough the rest of it is #

A point without a derivative can still have a definite roughness. The increment may scale like $|h|^\alpha$ for a single exponent $\alpha$, its Hölder exponent, and the exponent can differ from point to point. Here it does.

At $2\pi/3$ the slope is $0.50$: a square-root cusp, exponent $\tfrac12$. At $\pi$, once the $-h/2$ is removed, what is left shrinks like $|h|^{1.51}$ — the next term in the expansion is of order $|h|^{3/2}$, with a coefficient that oscillates without settling. At the irrational point the slope over four decades is $0.76$, but the curve is not a straight line. It rises in steps, steep where the window crosses a fraction whose Gauss sum is large and shallow where it does not, and a fit over a single decade gives anything from about $0.3$ to $1.3$.

Stéphane Jaffard computed the whole spectrum in 1996. At an irrational multiple of $\pi$, the exponent depends on how well the point is approximated by fractions whose Gauss sums do not vanish — fractions with $p$ or $q$ even — and for almost every point it is exactly $\tfrac34$. Points approximated unusually well by such fractions have exponents down to $\tfrac12$. So the function is rougher than a smooth function and smoother than a square root almost everywhere, and the roughness at a point is a statement about how close a fraction can get to that point. The golden ratio, the hardest number to approximate, sits at the typical value. The same kind of point-by-point exponent appeared for the devil’s staircase and its measure, where it was set by a self-similar rule; here it is set by arithmetic.

Where the derivative exists #

Sort every fraction with denominator up to 25 by what its Gauss sum says.

The derivative exists only in the top row. Those points are dense — between any two points of the period there are fractions with odd numerator and odd denominator — but they are countable, so they have measure zero, and everything between them, rational or irrational, has no derivative. Riemann’s function is differentiable on a set that meets every interval and occupies none of it.

The two middle rows are the one-sided cases. At those points one side of the Gauss sum’s combination vanishes and the other does not, so the function is flat with slope $-\tfrac12$ on one side and has a square-root cusp on the other. $\pi/2$ is the first of them. That such points exist at all is a surprise: a function with a genuine one-sided derivative on a dense set of points, and a cusp on the other side of each.

What the computation cannot establish #

A million terms and an exponent fitted over four decades establish the pattern. They do not prove it, and three things are beyond them. The first is the limit itself. A difference quotient that agrees with $-\tfrac12$ to two decimal places at $h = 3 \times 10^{-7}$ is evidence, and the derivative is a statement about $h \to 0$; Gerver’s proof is what makes it a theorem.

The second is the irrational points. The exponent $\tfrac34$ for almost every point is Jaffard’s theorem, and the measured $0.76$ is consistent with it. But a measurement at one irrational point cannot distinguish $\tfrac34$ from $0.7$ or $0.8$ when the local slope swings by a factor of four between decades, and the measurement says nothing about the exceptional points with smaller exponents, which exist but have measure zero.

The third is the role of the Gauss sum. The figure shows the coefficients matching the prediction at thirty-six fractions; the theta-function argument says why. The match at every fraction is a theorem, proved by the same modular identity, and no finite figure substitutes for it.

Still open: other series and the fine structure #

Riemann’s function is now one of the best-understood rough functions: its derivative, its exponents, its self-similarity and its multifractal spectrum are all known. Much of that is because theta is a modular form, and the knowledge does not transfer to series without that structure. For $\sum \sin(n^3 x)/n^3$, or for sums over the primes such as $\sum \sin(p x)/p$, the analogous questions — where is there a derivative, and what is the exponent at a typical point — are largely open, because no modular symmetry organises the phases. Sums with cubic phases lead into the theory of Weyl sums, where much less is known.

There is also a question about the complex version, $\sum e^{i n^2 x}/n^2$, whose real part is close to Riemann’s function and whose image in the plane is a curve. That curve turns out to describe the motion of a corner of a vortex filament — a thin tube of spinning fluid shaped initially as a polygon — in work by Francisco de la Hoz and Luis Vega. Whether the curve’s fractal dimension, its geometric self-similarity and its behaviour at irrational times can be computed in the same detail as the real part’s is an active question. So is whether any of it can be seen in a real fluid.

A candidate that nearly worked #

Riemann’s function was offered as a function with no derivative, and for nearly a century it was assumed to be one. It is instead a function whose derivative exists on a countable dense set of points and is the same number at every one of them. The number is $-\tfrac12$, left by a term that the series omits, and the points are exactly those where a sum of $2q$ points on the unit circle cancels to zero. The roughness everywhere else has three regimes — square roots at most fractions, three-quarter powers almost everywhere, and the exceptions between — and every one of them is read from how well a point can be approached by fractions whose Gauss sums do not vanish.

A curve with a corner at every point built its roughness by design, choosing ratios that guarantee a corner at every scale. Riemann’s function was not designed that way, and it shows what a natural series can do instead. It is smooth almost nowhere and differentiable exactly where number theory says, and Weierstrass’s search for a proof failed because there was nothing to prove.