Pair correlation of the zeros of the derivative of the Riemann \(\xi\)-function
The authors, assuming the Riemann hypothesis, investigate the pair correlation function \(F_1(\alpha,T)\) of the zeros of the derivative of the Riemann \(\xi\)-function \(\xi(s)\), \(s=\sigma+it\), defined by NEWLINE\[NEWLINE \xi(s)=\frac{1}{2}s(s-1)\pi^{-\pi/2}\Gamma(s/2)\zeta(s) NEWLINE\]NEWLINE NEWLINEIt is an entire function of order 1 satisfying the functional equation \(\xi(1-s)=\xi(s)\), and only has zeros identical to the complex zeros of the Riemann zeta-function \(\zeta(s)\).NEWLINENEWLINEThe authors prove that, for \(0<\delta<1\), NEWLINE\[NEWLINE\begin{multlined} F_1(\alpha,T)=N_1(T)^{-1}\sum_{0<\gamma, \;\gamma' \leq T}T^{i\alpha(\gamma-\gamma')}w(\gamma-\gamma')\\ =(1+o(1))T^{-2|\alpha|}\log T+|\alpha|-4|\alpha|^2+\sum_{k=1}^{\infty}\frac{(k-1)!}{(2k)!}(2|\alpha|)^{2k+1}+o(1), \end{multlined}NEWLINE\]NEWLINE uniformly for \(|\alpha|\leq 1-\delta\) as \(T \to \infty\); where the sum runs over pairs of ordinates of zeros of \(\xi'(s)\), \(w(u)=4/(4+u^2)\) is a weight function, the normalizing factor \(N_1(T)\sim (T/2\pi)\log T\) is the asymptotic number of zeros of \(\xi'\) with ordinates in \([0,T]\). Using this result, some corollaries on the number of zeros of the functions \(\xi(s)\) and \(\xi'(s)\) are obtained.NEWLINENEWLINEThe main result is an analogue to that one of \textit{H. L. Montgomery} on the pair correlation of the zeros of \(\xi(s)\) [in: Analytic Number Theory, Proc. Sympos. Pure Math. 24, St. Louis Univ. Missouri 1972, 181--193 (1973; Zbl 0268.10023)].
scientific article; zbMATH DE number 6327936
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