Exponential sums

An exponential sum is an expression of the form \[ \sum_{n=1}^N e^{2 \pi i f(n)},\] where \( f \) is a real-valued function defined on the positive integers. Such sums are used in the solution of various problems in number theory; in this article we will just play around with a few examples, draw their graphs and try to explain some of their features. By the "graph" of an exponential sum we mean the sequence of partial sums, plotted in the complex plane, with successive points joined by straight line segments. That is, we start at the origin; draw a line interval corresponding to the first term of the sum; from the end of this interval draw another, corresponding to the second term of the sum; and so on.

A good place to start is the exponential sum with \( f(n)=( \ln n)^4 \) and, say, \( N=5000 \). The graph was dubbed "the Loch Ness monster" by John Loxton in a 1981 article.

 

Graph of the exponential sum \( \sum_{n=1}^{5000} e^{2\pi i (\ln n )^4} \). Click on the image to enlarge.

To explain the shape of the graph we concentrate on the angle between successive line segments. Because of the factor \( 2 \pi \) in the exponential sum, we are measuring angles in units of full circles: that is, \( \frac14 \) represents a right angle, \( \frac12 \) represents a \( 180^\circ \) angle, and so on. The "blobs" in the Loch Ness monster occur when the angle between the \( n\)th line segment and the \( (n+1)\)th is close to an integer plus a half for several consecutive \( n \), so that the line segments (all of which have the same length) "fold back" upon one another. Correspondingly, the "smooth" parts correspond to values of \( n \) where the angle is close to an integer, so that the curve maintains substantially the same direction for a while.

We can understand this in more detail by noting that the angle between the \( n \)th and the \( (n+1) \)th line interval is \( f(n+1)-f(n) \), which is approximately the derivative \( f'(n) \). For the "monster" we have \[ f'(n)= \frac{4(\ln n)^3}{n} ; \] this tends to zero as \( n\to\infty \), and will at some stage take values of interest close to \( \frac12\), 1, \( \frac{3}{2} \) and so on. In fact, the "blob" at the bottom of the diagram is a result of the fact that \( f'(n) \approx \frac{1}{2} \) when \( n \) is about 4900; the previous one comes from \( f'(n) \approx \frac{3}{2} \) and so forth. The small blob at top left corresponds to \( f'(n)\approx \frac{11}{2} \). The almost vertical smooth section in the middle of the diagram is when \( f'(n) \approx1 \), the previous one \( f'(n) \approx2 \).

A less visually appealing example, but one which provides much food for thought, is given by \( f(n)= \frac{1}{2} \pi n^2 \).


Graph of the exponential sum \( \sum_{n=1}^{100} e^{ 2\pi i (\pi n^2 /2)} \). Click on the image to enlarge.

Since \( f'(n)=n \pi \), and since angles of integer size don't count, the behaviour of this graph is governed by the fractional parts (that is, the parts after the decimal point) of multiples of \( \pi \). The origin of this graph is near the bottom left of the diagram; by counting segments you can easily see that the section between the 3rd and 4th segments is almost straight. The graph appears to end near the middle of the diagram. In fact, this is not so: at this point, the 57th segment doubles back on the 56th so closely that you can't visually tell them apart. Further back towards the beginning, you can distinguish two nearby but separate paths. Two extremely accurate fractional approximations to \( \pi \) are \( \frac{22}{7} \) (well known) and \( \frac{355}{113} \) (less well known); if you add the pairs of numbers I have just mentioned you obtain \( 3+4=7 \) and \( 56+57=113 \), the denominators of these fractions. This is no coincidence: the numbers associated with significant features of the graph are not merely a topic for your next trivia night, but are profoundly connected with the value of \( \pi \).

Different types of functions \( f(n) \) give very different graphs and may not have any particularly noticeable "blobs" at all. If \( f \) is a polynomial the graphs often (though not always) display a beautiful symmetry, combined with a fascinating level of fine detail. Taking, for example, the cubic \( f(n)=n/11+n^2/21+n^3/31 \), we obtain the following delightful graph.



Graph of the exponential sum
\( \sum_{n=1}^{7161} e^{2\pi i (n/11 + n^2/21 + n^3/31) } \). Click on the image to enlarge.

You can even personalise the graphs by choosing \[ f(n)= \frac{n}{dd} + \frac{n^2}{mm} + \frac{n^3}{yy} , \] where \( dd.mm.yy \) is your own date of birth. Here's mine: if you can deduce the values of \( dd, mm \) and \( yy \), don't forget to send me a birthday card.

 

Graph of the exponential sum \( \sum_{n=1}^{10620} e^{2\pi i f(n)} \). Click on the image to enlarge.

"Real-life" applications of these ideas? Forget it. This is mathematics for fun. Nothing wrong with that.

David Angell, School of Mathematics and Statistics, UNSW.

Further information
J.H. Loxton, The Graphs of Exponential Sums. Mathematika 30 (1983), 153--163.