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On the distribution of spacings between zeros of the zeta function. (English) Zbl 0615.10049
The Riemann Hypothesis (RH) is the most famous unsolved problem in number theory, and perhaps even in the whole of mathematics. Although more than a thousand hours have been spent on a modern supercomputer to verify that the first $1\cdot 5\times 10\sp 9$ zeros do lie on the half line [{\it J. van de Lune}, {\it H. J. J. te Riele} and {\it D. T. Winter}, ibid. 46, 667-681 (1986; Zbl 0585.10023)] the proof of RH is still as elusive as it has ever been. Nevertheless many number theorists not only believe that RH is true but made further speculations on problems such as the magnitude of the largest prime gaps, answers to which lie well beyond our current knowledge of the zeta function. In 1973 {\it H. L. Montgomery} [Proc. Symp. Pure Math. 24, 181-193 (1973; Zbl 0268.10023)] put forward his pair correlation conjecture. Roughly speaking this says that not only all the zeros are on the half line, but they have 1-(sin $\pi$ u/$\pi$ u)${}\sp 2$ as their pair correlation function. Meanwhile it was pointed out by F. J. Dyson that the Gaussian unitary ensemble (GUE), which has been studied extensively in mathematical physics as a model for distribution of energy levels in many-particle systems, has the same pair correlation function. The suggestion then is that there is a connection between the zeros of the zeta function and the eigenvalues of random Hermitian matrices. This very interesting paper reports on the numerical study of the distribution of the spacing between zeros of the zeta function. The normalized spacing between consecutive zeros $+i\gamma\sb n$ and $+i\gamma\sb{n+1}$ is given by $$\delta\sb n=(\gamma\sb{n+1}-\gamma\sb n)\frac{\log (\gamma\sb n/2\pi)}{2\pi}$$ and many graphs and tables are presented. Empirical data on pair correlation, probability density, and various statistical tests are given and compared to results obtained from the GUE predictions. There does seem to be general agreement with the predictions especially at zeros with large height. Where there is a pronounced disagreement the author explains the phenomenon by relating it to the prime numbers. For example, there are fewer very large and very small $\delta\sb n$ and more $\delta\sb n$ near the mean value of 1 than predicted, and an explanation is given in terms of the size of $S(t)=(1/\pi)\arg \zeta (1/2+it)$. It is also reported that the largest $\delta\sb n$ found is $\delta\sb n=4\cdot 2626...$, for $n=184 155 671 040$ which was located when in investigating the violations of Rosser’s rule. The samples are drawn from blocks of $10\sp 5$ consecutive zeros $1/2+i\gamma\sb n$ at $n=1$, $10\sp 9$, $10\sp{11}$, $2\times 10\sp{11}$ and $10\sp{12}$, and there is a godd description on the method of computation explaining the difficulty involved in obtaining the accuracy of $\pm 10\sp{-8}$ onthe Cray-1 and Cray X-MP computers. As has happened before, when powerful machines are pushed to their limits by careful experts, bugs in software are discovered during computations. Thus accuracy is lost on calculations corresponding to $10\sp 9$, $10\sp{11}$ and $2\times 10\sp{11}$ so that only two sets of data are presented. Since the two subjects concerned are apparently unrelated the problems arising are naturally intractable. Although no firm conclusion can be drawn from the paper it will encourage workers to seek a connection. Moreover, if the link is genuine and the complexity of the relationship can be revealed, then the gain to science will be unparalleled.
Reviewer: P.Shiu

##### MSC:
 11M06 $\zeta (s)$ and $L(s, \chi)$
BRENT
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