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The \(n\)-level correlations of zeros of the zeta function. (English. Abridged French version) Zbl 0837.11047

Let \(\gamma_1 \leq \gamma_2 \leq \dots\) run over the imaginary parts of the zeros of the zeta-function in the upper half-plane; then the numbers \(\widetilde \gamma_j = (\gamma_j \log \gamma_j)/2 \pi\) have mean spacing unity. The authors consider vectors \((\widetilde \gamma_{j_1} - \widetilde \gamma_{j_2}, \dots, \widetilde\gamma_{j_{n-1}}-\widetilde \gamma_{j_n})\) related to \(n\)-tuples \(S = (\widetilde \gamma_{j_1}, \dots, \widetilde \gamma_{j_n})\), where the indices \(j_i\) are distinct and do not exceed a given bound \(N\). The distribution of these vectors may be characterized in terms of sums \((n!/N) \sum_S f(\widetilde \gamma_{j_1}, \dots, \widetilde \gamma_{j_n})\), where \(S\) runs over the \(n\)-tuples mentioned above, and \(f\) is a “test function” of a certain type. If, in addition, the Fourier transform \(\widehat f(\xi)\) of \(f\) is supported in \(\sum_j |\xi_j |< 2\), then an asymptotic formula for this sum is given as \(N \to \infty\), under the assumption of the Riemann hypothesis. This result generalizes a well- known theorem of H. L. Montgomery [Proc. Sympos. Pure Math. 24, 181-193 (1973; Zbl 0268.10023)] on the “pair correlation” for the zeros of the zeta-function, and supports a generalization of his pair correlation conjecture. The argument of the authors makes sense also unconditionally.
Reviewer: M.Jutila (Turku)

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0268.10023
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