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Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function. (English) Zbl 1362.11078

The authors study extensively the function \[ N(T,\beta) := \sum_{0<\gamma,\gamma'\leq T, 0<\gamma'-\gamma\leq \frac{2\pi\beta}{\log T}}1. \] Throughout the paper the Riemann hypothesis (RH) is assumed, i.e., all complex zeros of the Riemann zeta-function \(\zeta(s)\) have real parts equal to \(\frac12\). In the above formula \(\frac12+i\gamma\) and \(\frac12+i\gamma'\) denote zeros of \(\zeta(s)\) counted with multiplicity, and \(\beta>0\). The pair correlation conjecture of H. L. Montgomery [in: Analytic Number Theory, Proc. Sympos. Pure Math. 24, St. Louis Univ. Missouri 1972, 181–193 (1973; Zbl 0268.10023)] asserts that \[ N(T,\beta) \;\sim\; N(T)\int_0^\beta \left\{1 - \left(\frac{\sin\pi x}{\pi x}\right)^2\right\}{\mathrm d}x \qquad(\text{RH}, \beta>0, T\to\infty). \] Here \(N(T)\) denotes the number of zeros \(\frac12 + i\gamma\) for which \(0 < \gamma \leq T\). Since \(N(T)\) is asymptotic to \((T\log T)/(2\pi)\), it follows that the average size of \(\gamma_{n+1}-\gamma_n\) is about \((2\pi)/(\log \gamma_n)\), where \(0 < \gamma_1 \leq \gamma_2 \leq \cdots\) denote ordinates of zeta-zeros. Hence \(N(T,\beta)\) counts the number of pairs of zeta ordinates whose difference does not exceed \(\beta\) times the average spacing. The authors prove various interesting upper and lower bounds for \(N(T,\beta)\), using Montgomery’s formula and some extremal functions of the exponential type. The functions in question are optimal in the sense that they majorize and minorize the characteristic function of the interval \([-\beta, \beta]\) in order to minimize a certain \(L^1\)-norm. The paper contains a complete solution of this extremal problem using Hilbert spaces of entire functions. This extends the work of P. X. Gallagher [J. Reine Angew. Math. 362, 72–86 (1985; Zbl 0565.10033)]. To illustrate their results, we present here just two of the many results proved in the paper: If \(\beta\left(\frac{\log\log T}{\log T}\right)^{1/2}\to0\) as \(T\to\infty\), then under the RH \[ \beta - \frac76 + \frac{1}{2\pi\beta^2} + O(\beta^{-2}) + o(1) \leq \frac{N(T,\beta)}{N(T)} \leq \beta + \frac{1}{2\pi\beta^2}+ O(\beta^{-2}) + o(1), \] and \(N(T,0.606894) \gg N(T)\) if almost all zeta-zeros are simple and the RH holds.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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