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The Spectrum of Riemannium

Brian Hayes

Eulerium and Riemannium

So much for nuclear physics; what about number theory and the zeta function?

The most celebrated sequence in number theory is that of the primes: 2, 3, 5, 7, 11.... The overall trend in this series is well known. In the neighborhood of any large integer x, the proportion of numbers that are prime is approximately 1/log x, which implies that although the primes go on forever, they get sparser as you climb farther out on the number line. Superimposed on this gradual thinning of the crop are smaller-scale fluctuations that are harder to understand in detail. The sequence of primes looks quite random and erratic, and yet it cannot possibly have the same nearest-neighbor statistics as a truly random spectrum. The nearest that two primes can approach each other (except in one anomalous case) is 2. Pairs that have this minimum spacing, such as 29 and 31, are called twin primes. No one knows whether there are infinitely many of them.

In addition to directly exploring the primes, mathematicians have taken a roundabout approach to understanding their distribution by way of the Riemann zeta function. This function, although named both by and for Bernhard Riemann, was first studied in the 18th century by Leonhard Euler, who defined it as a sum over all the natural numbers:

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In other words, take each natural number n from 1 to infinity, raise it to the power s, take the reciprocal, and add up the entire series. The sum is finite whenever s is greater than 1. For example, Euler showed that z(2) is equal to p2/6, or about 1.645:

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Euler also proved a remarkable identity, equating the summation formula, with its one term for each natural number, to a product formula that has one term for each prime. This second definition states:

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The recipe in this case is to take each prime p from 2 to infinity, raise it to the power s, then after some further arithmetic multiply together the terms for all p. The result is the same as that of the summation. This connection between a sum over all integers and a product over all primes was a hint that the zeta function might have something to say about the distribution of primes among the integers, and in fact the two series are intimately related.

Figure 4. Zeros of theClick to Enlarge Image

Riemann's contribution, in 1859, was to extend the domain of the zeta function so that it applies not just when s is a real number greater than 1 but when s is any number—positive or negative, real or complex—with the single exception of numbers whose real part is equal to 1. Over much of the complex plane the function turns out to be wildly oscillatory, crossing from positive to negative values infinitely often. The crossing points, where ζ(s)=0, are called the zeros of the zeta function. There is an infinite series of them along the negative real axis, but these are not looked upon with great interest. Riemann called attention to a different infinite series of zeros lying above and below the real axis in a vertical strip of the complex plane that includes all numbers whose real part is between 0 and 1. Riemann calculated the locations of the first three of these zeros and found that they lie right in the middle of the strip, on the "critical line" with real coordinate 1/2. On the basis of this evidence, plus incredible intuition, he conjectured that all the complex zeros are on the critical line. This is the Riemann hypothesis, widely regarded as the juiciest prize plum in all of contemporary mathematics.

In the years since Riemann located the first three zeta zeros, quite a few more have been found. A cooperative computing network called ZetaGrid, organized by Sebastian Wedeniwski of IBM, has checked 385 billion of them. So far, every one is on the critical line. There's even a proof that infinitely many lie on the line, but what's wanted is a proof that none lie anywhere else. That goal remains out of reach.

In the meantime, other aspects of the zeta zeros have come under scrutiny. Assuming that all the zeros are indeed on the critical line, what is their distribution along that line? How does their density vary as a function of the "height," T, above or below the real axis?

As with the primes, the overall trend in the abundance of zeta zeros is known. The trend goes the opposite way: Whereas primes get rarer as they get larger, the zeta zeros crowd together with increasing height. The number of zeros in the neighborhood of height T is proportional to log T, signifying a slow increase. But, again, the trend is not smooth, and the details of the fluctuations are all-important. Gaps and clumps in the series of zeta zeros encode information about corresponding features in the sequence of primes.

Montgomery's work on the pair-correlation function of the zeta zeros was a major step toward understanding the statistics of the fluctuations. And the encounter in Fuld Hall, when it emerged that Montgomery's correlation formula is the same as that for eigenvalues of random matrices, ignited further interest. The correlation function implies level repulsion among the zeros just as it does in the nucleus, producing a deficiency of closely spaced zeros.

Figure 5. Pair-correlationClick to Enlarge Image

Montgomery's result is not a theorem; his proof of it is contingent on the truth of the Riemann Hypothesis. But the accuracy of the correlation function can be tested by comparing the theoretical prediction with computed values of zeta zeros. Over the past 20 years Andrew M. Odlyzko, now at the University of Minnesota, has taken the computation of zeta zeros to heroic heights in order to perform such tests. For this purpose it is not enough to verify that the zeros lie on the critical line; the program must accurately measure the height of each zero along that line, which is a more demanding task. One of Odlyzko's early papers was titled "The 1020th zero of the Riemann zeta function and 175 million of its neighbors." Since then he has gone on to compute even longer series of consecutive zeros at even greater heights, now reaching the neighborhood of the 1023rd zero. The agreement between predicted and measured correlations is striking, and it gets better and better with increasing height.

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