Triple correlation of the Riemann zeros (Q1011972)

The goal of this article is to give a precise formula for the distribution of triplets of zeros of the Riemann zeta function, including lower-order terms. Under the Riemann Hypothesis (RH), an asymptotic formula is known for this statistic [\textit{D. A. Hejhal}, ``On the triple correlation of zeros of the zeta function'', Int. Math. Res. Not. 1994, No. 7, 293--302 (1994; Zbl 0813.11048)], where the \(\gamma_i\) represent the ordinates of the zeros of \(\zeta(s)\): \[ \begin{multlined} \sum_{\substack{ T\leq\gamma_1,\gamma_2,\gamma_3 \leq 2T \\ \gamma_i \text{ distinct}}} w(\gamma_1,\gamma_2,\gamma_3)f\left(\frac{\log T}{2\pi}(\gamma_1-\gamma_2),\frac{\log T}{2\pi}(\gamma_1-\gamma_3)\right) \\ \sim \frac{T\log T}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(u,v) \begin{vmatrix} 1 & S(u) & S(v) \\ S(u) & 1 & S(u-v) \\ S(v) & S(u-v) & 1 \end{vmatrix} \,du\,dv. \end{multlined}\tag{1} \] Here, \(w(\gamma_1,\gamma_2,\gamma_3)\) is a certain weight, \(S(x):=\frac{\sin(\pi x)}{\pi x}\), and the function \(f(u,v)\) and its Fourier transform satisfy some conditions which we do not state here. In the current article, the following formula is proven (under RH): \[ \begin{multlined} \sum_{\substack{ 0<\gamma_1,\gamma_2,\gamma_3<T \\ \gamma_i \text{ distinct}}} f(\gamma_1-\gamma_2,\gamma_1-\gamma_3) = \frac 1{(2\pi)^3} \int_{-T}^T \int_{-T}^T f(v_1,v_2) \\ \times \bigg[\int_0^T \log^3 \frac u{2\pi} \,du+I(iv_1,iv_2;0)+I(0,iv_1;-iv_2) +I(0,iv_2;-iv_1) \\ +I(-iv_1,-iv_2;0)+I(0,-iv_2;iv_1)+I(0,-iv_1;iv_2)+I_1(0;iv_2)+I_1(0;iv_1) \\ +I_1(-iv_2;iv_1)+I_1(-iv_2;0)+I_1(-iv_1;iv_2)+I_1(-iv_1;0)\bigg] \,dv_1\,dv_2+O_{\varepsilon}(T^{\varepsilon}), \end{multlined}\tag{2} \] where \[ I(\alpha_1,\alpha_2;\beta):=\int_0^T \frac{\zeta'}{\zeta}\left( \frac 12+it+\alpha_1\right)\frac{\zeta'}{\zeta}\left(\frac 12+it+\alpha_2\right)\frac{\zeta'}{\zeta}\left(\frac 12-it+\beta\right)\,dt, \] \[ I(\alpha;\beta):=\int_0^T \log \left(\frac t{2\pi}\right) \frac{\zeta'}{\zeta}\left(\frac 12+it+\alpha\right)\frac{\zeta'}{\zeta}\left(\frac 12-it+\beta\right)\,dt. \] Assuming the ``ratios conjecture'' of [\textit{B. Conrey, D. W. Farmer} and \textit{M. R. Zirnbauer}, ``Autocorrelation of ratios of L-functions'', Commun. Number Theory Phys. 2, No. 3, 593--636 (2008; Zbl 1178.11056)], the authors show how one can transform the terms involving \(I(\alpha_1,\alpha_2;\beta)\) and \(I_1(\alpha;\beta)\) into expressions involving values of the zeta function one the line \(\text{Re}(s)=1\). Using this, it can be seen that the formula (2) gives the main term as in (1) (without any restriction on the support of the Fourier transform of \(f(u,v)\)), but also includes interesting lower-order terms, up to an error of \(O_{\varepsilon}(T^{\frac 12+\varepsilon})\). This fact is far from being trivial, and a section of the paper is devoted to this question, in which the authors show how to recover (1) from (2) in a very clean way. Moreover, they discuss the reason why they did not use the usual normalization of multiplying the zeros by \(\frac{\log T}{2\pi}\) in (2), since it turns out the formula is much nicer without this normalization. Of course we can always take \(f_T(u,v):= g\left(\frac{\log T}{2\pi}u,\frac{\log T}{2\pi}v\right)\) to renormalize. The last section of the paper is devoted to the three-point statistics of eigenvalues of large unitary matrices, which is the random matrix theory counterpart of the zeta zeros statistics. The authors derive a formula analogous to (2) including lower-order terms. This is done using the ``ratios theorem'', hence the result is unconditional. Two distinct proofs are given. The first one is very simple and uses a lemma of Gaudin. The second one is more involved, however it is using ideas analogous to that in the proof of (2), highlighting the similarities of the two questions, thus strengthening the links between random matrices and zeta zeros. Finally, the authors provide numerical calculations throughout the article, which are very well placed along the formulas to highlight the importance of each of the terms they contain. A striking feature of these pictures is the appearance of vertical, horizontal and oblique lines appearing at each zero of the zeta function, and this phenomenon is very well explained by the occurrence of logarithmic derivatives of the zeta function in the lower-order terms of (2).