prime numbers, the zeta function and li

Firstly, this page has nothing to do with the logarithmic integral function Li(x) which is defined by \[\mathrm{Li}(x)=\int_{2}^{x}\frac{du}{\log u}\] and which approximates the number of primes < x.

Similarly, it has nothing to do with the mathematician Xian-Jin Li who has reformulated the Riemann hypothesis.

It is about an ancient Chinese concept called li and its possible relevance to our understanding of the distribution of primes.
 

I first came across the concept in 1991, in this excerpt from p.15 of Alan Watts' beautiful book Tao: The Watercourse Way (Pantheon, 1975):

"I have practiced Chinese calligraphy for many years, and am not yet a master of the art, which could be described as dancing with brush and ink on absorbent paper. Because ink is mostly water, Chinese calligraphy - controlling the flow of water with the soft brush as distinct from the hard pen - requires that you go with the flow. If you hesitate, hold the brush too long in one place, or hurry, or try to correct what you have written, the blemishes are all too obvious. But if you write well there is at the same time the sensation that the work is happening on it own, that the brush is writing all by itself - as a river, by following the line of least resistance, makes elegant curves. The beauty of Chinese calligraphy is thus the same beauty which we recognize in moving water, in foam, spray, eddies, and waves, as well as in clouds, flames and weavings of smoke in sunlight. The Chinese call this kind of beauty the following of li, an ideogram which referred originally to the grain in jade and wood, and which Needham translates as "organic pattern," although it is more generally understood as the "reason" or "principle" of things. Li is the pattern of behaviour which comes about when one is in accord with the Tao, the watercourse of nature. The patterns of moving air are of the same character, and so the Chinese idea of elegance is expressed as feng-liu, the flowing of wind."

This is from another essay by A. Watts on Taoism:

"The tao is a certain kind of order, and this kind of order is not quite what we call order when we arrange everything geometrically in boxes, or in rows. That is a very crude kind of order, but when you look at a plant it is perfectly obvious that the plant has order. We recognize at once that it is not a mess, but it is not symmetrical and it is not geometrical looking. The plant looks like a Chinese drawing, because they appreciated this kind of non-symmetrical order so much that it became an integral aspect of their painting. In the Chinese language this is called li, and the character for li means the markings in jade. It also means the grain in wood and the fiber in muscle. We could say, too, that clouds have li, marble has li, the human body has li. We all recognize it, and the artist copies it whether he is a landscape painter, a portrait painter, an abstract painter, or a non-objective painter. They all are trying to express the essence of li. The interesting thing is, that although we all know what it is, there is no way of defining it. Because tao is the course, we can also call li the watercourse, and the patterns of li are also the patterns of flowing water. We see those patterns of flow memorialized, as it were, as sculpture in the grain in wood, which is the flow of sap, in marble, in bones, in muscles. All these things are patterned according to the basic principles of flow. In the patterns of flowing water you will find all kind of motifs from Chinese art, immediately recognizable, including the S-curve in the circle of yang-yin.

So li means then the order of flow, the wonderful dancing pattern of liquid, because Lao-tzu likens tao to water:

The great tao flows everywhere, to the left and to the right, It loves and nourishes all things, but does not lord it over them."
 

At the end of the 1990's, I was spontaneously struck by an extraordinary mental image – that of a 'dynamical history' underlying the familiar distribution of prime numbers. The mental image seemed to imply a sort of evolutionary process in which the prime numbers were seen to be moving particles in a 1-dimensional continuum, eventually coming to rest when they collectively achieved the dynamical equilibrium of their familiar configuration.

In the following years, while attempting to write about certain (more widely acknowledged) mysteries associated with the prime numbers and Riemann zeta function, and in particular about the sort of aesthetic reactions people tend to have to them, I invoked the notion of li. As I then understood it, it was typified by the kind of beauty associated with mists, clouds, smoke, rugged mountain ranges, gnarled trees, etc.

In 2003, Wooden Books (who published my little reference book) brought out a book called Li: Dynamic Form in Nature by architect David Wade. In the introduction he explains:

"Li are essentially dynamic formations, and as such can give the impression of a frozen moment, of a process caught at a particular instant of time..."

Wade's section on 'brechiated' li observes that