Correlations of zeros

The $n$-correlation functions of the zeros of the Riemann $\zeta$-function have been determined for a restricted class of test functions. See [49 #2590][ MR 96d:11093][ MR 97f:11074]. These results are established by relating the correlation functions to a sum over the prime numbers, and at present it is possible to prove a rigorous result only in the range where the ``diagonal terms'' in the sum are dominant.

Extending to a larger range would require some sort of information on sums of the form $\sum\Lambda(n)\Lambda(n+k)$, where $\Lambda$ is the Von Mangoldt function defined by $\Lambda(n)=\log p$ if $n=p^k$, $p$ prime, and $\Lambda(n)=0$ otherwise. These sums appear to be closely related to the ``twin prime'' problem, because $\Lambda(n)\Lambda(n+k)$ is nonzero only when $n$ and $n+k$ are both primes (or prime powers, which is not a significant contribution). Bogolmony and Keating see [Nonlinearity 8, 1115-1131] and [Nonlinearity 9, 911-935], derive all $n$-correlation functions by assuming the Hardy-Littlewood conjectures and ignoring error terms, and the result agrees with the GUE conjecture. The calculation involves some difficult combinatorics.

The Hardy-Littlewood twin prime conjectures are too strong of an input into this problem, because it is averages of sums of the form $\sum\Lambda(n)\Lambda(n+k)$ which need to be evaluated, and the information about primes of a specific form is lost in the averaging. In particular, the GUE hypothesis does not imply the Hardy-Littlewood conjectures. Goldston, Gonek, and Montgomery have shown that the pair correlation conjecture is equivalent to a statement about the variation of the distribution of primes. This is not even strong enough to imply that there is a $k$ for which $p_{n+1}-p_n=k$ infinitely often. It appears that GUE hypothesis for $n$-correlation is equivalent to a statement about the variation in the distribution of $n-1$-almost primes.

It would be a significant accomplishment to prove anything about the correlation functions outside the range in which they currently are known. Two results in this direction are Özlük's work [ MR 92j:11091] on the $Q$ aspect of pair correlation for Dirichlet L-functions, and recent work of Goldston, Gonek, Özlük and Snyder [ MR 2000k:11100] in which they prove a lower bound for $F(\alpha;T)$ for $1<\alpha<\frac32$.

It also would be valuable to have an idea, assuming GUE, of the rate at which the $n$-correlation sums converge to their limiting behavior, and to have an idea of how that rate changes as $n\to\infty$. See ratios of zeta-functions for some additional discussion.

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